UC-NRLF, 


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Z61 
M33 


jM*B   531   b77 

THE  PRINCIPLES 


OP 


ELLIPTIC    AND    HYPERBOLIC 

ANALYSIS 


BY 


ALEXANDER   MACFARLANE,   M.A.,  D.Sc,  LL.D. 

Fellow  of  the  Koyal  Society  of  Edinburgh  ;  Professor  ok  Physics 

IN    THE    ITnIVERSJTY    OF    TfXAS 


>>«<€ 


J.    S.   CUSHING  &   CO.,  PRINTERS 

BOSTON,    MASS.,   U.S.A. 


GIFT   OF 


GIFT  or 


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UHIVBRSrTT; 


THE  PRINCIPLES 


OP 


ELLIPTIC    AND    HYPERBOLIC 
ANALYSIS 


BY 


ALEXANDER   MACFARLANE,  M.A.,  D.Sc,  LL.D. 

Fellow  of  the  Koyal  Society  of  Edinburgh  ;  Professor  of  Physics 
IN  THE  University  of  Texas 


fTJSIVBESIT 


3»i< 


J.   S.   GUSHING  &  CO.,  PRINTERS 
BOSTON,  MASS.,  U.S.A. 


M33 


Copyright,  1894, 
By  ALEXANDER  MACFARLANE. 


THE  PRINCIPLES   OF   ELLIPTIC   AND 
HYPERBOLIC  ANALYSIS. 


[Abstract  read  before  the  Mathematical  Congress  at  Chicago, 
August  24,  1893.*] 


Ix  several  papers  recently  published,  entitled  ''Principles  of 
the  Algebra  of  Physics,"  "  The  Imaginary  of  Algebra,"  and  ''  The 
Fundamental  Theorems  of  Analysis  generalized  for  Space,"  I  have 
considered  the  principles  of  vector  analysis  ;  and  also  the  princi- 
ples of  versor  analysis,  the  versor  being  circular,  logarithmic,  or 
equilateral-hyperbolic.  In  the  present  paper,  I  propose  to  con- 
sider the  versor  part  of  space  analysis  more  fully,  and  to  extend 
the  investigation  to  elliptic  and  hyperbolic  versors.  The  order 
of  the  investigation  is  as  follows  :  The  fundamental  theorem  of 
trigonometry  is  investigated  for  the  sphere,  the  ellipsoid  of  revo- 
lution, and  the  general  ellipsoid ;  then  for  the  equilateral  hyper- 
boloid  of  two  sheets,  the  equilateral  hyperboloid  of  one  sheet, 
and  the  general  hyperboloid.  Subsequently,  the  principles  arrived 
at  are  applied  to  find  the  complete  form  of  other  theorems  in 
spherical  trigonometry,  and  to  deduce  the  generalized  theorems 
for  the  ellipsoid  and  the  hyperboloid.  At  the  end,  the  analogues 
of  the  rotation  theorem  are  deduced. 


FUNDAMENTAL  THEOREM  FOR  THE  SPHERE. 

Let  a^  and  jB^  denote  any  two  spherical  versors  ;  their  planes 
will  intersect  in  the  axis  which  is  perpendicular  to  a  and  (S,  and 

*  Jan.  8,  1894.  I  have  rewritten  and  extended  the  original  paper  so  as  to 
include  the  trigonometry  of  the  general  ellipsoid  and  hyperboloid.  At  the 
time  of  reading  the  paper,  I  had  discovered  how  to  make  this  extension,  but 
had  not  had  time  to  work  it  out. 

1 


368655 


2       PRINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC  ANALYSIS. 

Which  we  denote  by  ^.  Let  OFA  (Fig.  1)  represent  <  and 
OAQ  represent  ^^;  then  OPQ,  the  third  side  of  the  spherical  tri- 
angle, represents  the  product  a^yg*. 

To  prove  that 

a^(B^  =  cos  ^  cos  5  -  sin  ^  sin  5  cos  aft 

-f  Jcos5sin^.«  +  cos^sin^.yg_sin^sinJBsina^.cc/8jl 

The  first  part  of  this  proposition,  namely,  that 

cos  a^ft^  =  cos  ^  cos  ^  -  sin  A  sin  B  cos  aft, 

is  equivalent  to  the  well-known  fundamental  theorem  of  Spherical 

Trigonometry ;  the  only  difference  is, 
that  a;8  denotes,  not  the  angle  included 
by  the  sides,  but  the  angle  between 
the  planes;  or,  to  speak  more  accu- 
rately, the  angle  between  the  axes  a 
and  ft.  It  is  more  difficult  to  prove  the 
complementary  proposition,  namely, 
that 

Sin  a''ft^=  cos  -B  sin  ^ .  a  -f-  cos  A  sin  B  -  ft 
—  sin  A  sin  B  sin  aft  •  aft, 
for  it  is  necessary  to  prove,  not  only  that  the  magnitude  of  the 
right-hand  member  is  equal  to  Vl^^^'^^^^-,  but  also  that  its  ' 
direction  coincides  with  the  axis  normal  to  the  plane  of  OPQ 
At  page  7  of  "Fundamental  Theorems,'^  I  have  proved  the  above 
statement  as  regards  the  magnitude,  but  I  was  then  unable  to 
give  a  general  proof  as  regards  the  axis.     Now,  however,  I  am 
able  to  supply  a  general  proof,  and  it  will  be  found  of  the  highest 
importance  in  the  further  development  of  the  analysis. 

/r/f  \  ^^ '"  *^^  "''^'^^  ^"'^  ^^  <  ^^^  ^e  the  terminal  line 
of  ft^',  let  OF  be  drawn  equal  to 

cos5sin^.«4-cos^sini?.^_sin^sinZ?sin«^.«^; 
it  is  required  to  prove  that  OF  is  perpendicular  to  OF  and  to 

Now,  OP=a-^^  =  (cosA-smA.J).o^ 


PRINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC   ANALYSIS.      3 


Similarly,    OQ^^^a^    =  (cos  B -\- sin  B- I3^)'ap 

=  G0sB'aP  +  smB'(3^  a^. 

By  a^  ap  is  meant  the  axis  which  is  perpendicular  to  a  and  (3, 
after  it  is  rotated  by  a  quadrant  round  a.     In  Fig.  2,  let  OA  and 
OB  represent  a  and  jS,  any  two  axes 
drawn  from  0,  then  a(3  is  drawn  from 
O  upwards,  normal  to  the  plane  of  the 

w   

paper.  Hence  a^  a^  is  OL,  which  is 
of  unit  length,  and  drawn  in  the  plane 
of  the  paper,  perpendicular  to  a.  It 
is  required  to  find  the  components  of 
OL  along  a  and  /8.  Draw  LN  par- 
allel   to    /8,    and    LM  parallel  to  a. 

Kow  OM  or  NL  is -. •  13,   and 


OiV^  is 


cos  aft 
sin  a/? 


•  a ;  hence, 


sin  ajS 


I  —5      cos  a3 
sm  afS 


sin  aft 


/3- 


Similarly,  /3' «/S  =  -/3"/8« 


I^_     cosojS 


/8  + 


a. 


sin  a(3  '     '  sina^ 

Consequently,  the  three  lines  expressed  in  terms  of  the  axes  a, 
p,  and  a^,  are 

OR  =      cos  jB  sin  ^  .  a  -f  cos  A  sin  B  •  (3  —  sin  A  sin  Bsina^-a^-, 


0P  =  -  sin  A 


OQ  =      sin  B 


COS  al3 
sin  ap 

1 


ct  -f  sin  ^ 


sina^ 

•     TtCos  aB 

-a  —  smB ^ 

sm  a/3 


(3 -^  cos  A  •  a/3 ; 
/3-\-cosB'ap. 


sin«^ 
Hence     cos  (01^)  (OP) 

=  -  cos  J5  sin^^^^^i^  -  ^-^^ 
ysm  a^     sm  a^J 


cos  A  sin  A  sin  b[^-^^ ^ 

y  sin  aft      sin  «y8 


sm 


«iS) 


=  0. 


Similarly,  it  may  be  shown  that  cos(OR){OQ)  —  0y  hence  OR 
has  the  direction  of  the  normal  to  the  plane  of  OPQ. 


PRINCIPLES  OF  ELLIPTIC   AND  HYPERBOLIC  ANALYSIS. 


To  find  the  general  expression  for  a  spherical  versor,  when  refer- 
ence is  made  to  a  principal  axis. 

Let  OA  represent  the  principal  axis  (Fig.  3),  and  let  it  be 
denoted  by  a.  Any  versor  OPA,  which  passes  through  the  prin- 
cipal axis,  may  be  denoted  by  /8",  where  ^  denotes  a  unit  axis 
perpendicular  to  a.  Similarly,  OAQ,  another  versor  passing 
through  the  principal  axis,  may  be  denoted  by  y",  where  y  denotes 
a  unit  axis  perpendicular  to  a.  The  product  versor  OPQ  is  circu- 
lar, but  it  will  not,  in  general,  pass  through  OA ;  let  it  be  denoted 
by  ^^.     Now 

^^  =  ^7" 

=  cos  u  cos  V  —  sin  ?f  sin  v  cos  ^y 


-f  Jcos-u  sinw-^4-  coswsiny-y  —  sinwsinvsiujSy  ./Syj 


We  observe  that  the  directed  sine  may  be  broken  up  into 
two  components,  namely,  cos  v  sin  u-  /S  -\-  cos  u  sin  v  •  y,  which  is 
perpendicular  to  the  principal  axis,  and    —  sin  u  sin  v  sin  jSy  jSy^ 

which  has  the  direction  of  the 
negative  of  the  principal  axis,  for 
py=a. 

Draw  OS  to  represent  the  first 
component  cosvsinii'  (3,  OT  to 
represent  the  second  component 
cosiisin'y-y,  and  OU  to  represent 
the  third  component  —cosugosv 
sinpya.  Draw  OV,  the  resultant 
of  the  first  two,  and  OR,  the  re- 
sultant of  all  three.  The  plane 
of  OA  and  OV  passes  through 
OR,  which  is  normal  to  the  plane 
OPQ ;  hence  these  planes  cut  at  right  angles  in  a  line  OX ;  and 
the  angle  between  OA  and  OX  is  equal  to  that  between  OV  and 
OR,  for  OF  is  perpendicular  to  OA,  and  OR  to  OX.  Let  <^ 
denote  the  angle  AOX,  then 


cos  <f»  =  "V^^QS^'?;  sin^u + cos^u  sin^u  -f-  2  cos  ^  cos  v  sin  u  sin  v  cos  /8y 


and 

sin  <^ 


Vl  —  (cos  u  cos  V  —  sin  u  sin  v  cos  /8y ) ' 
sin  ?^  sin'?;  sin /8y 


Vl— (cos  I*  cosy  —  sin  w  sin  V  cos  ^y)^ 


PRINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC  ANALYSIS.       O 

Figure  4  represents  a  section  through  the  plane  of  OA  and  0  V. 
Let  XM  be  drawn  from  X  perpendicular  to  OA-,  it  is  equal  in 
magnitude  to  sin  <^ ;  and  OM  is  equal  in 
magnitude  to  cos  <f>. 

Hence  the  axis  $  has  the  form 

cos  <^  •  €  —  sin  <f> '  a, 

where  c  denotes  a  unit  axis  perpendicular 
to  a.     And 

^^  ==  cos  0  -f  sin  ^(cos  <{>  •  e  —  sin  <f>  >  a)^ 

is  determined  by  the  equations, 

cos  6  =  cos  u  cos  V  —  sinu  sin  v  cos  jSy,  (1) 

sin  0  sin  <^  =  sin  u  sin  v  sin  f3y,  (2) 

sin  0  cos  cf>-e  =  cos  v  sin u- ^  -\-  cos  it  sin  v  •  y.  (3) 

The  unit  axis  e  may  be  expressed  in  terms  of  two  axes  p  and  y, 
which  are  at  right  angles  to  one  another  and  to  a,  and  the  angle 
which  c  makes  with  /3.  Hence  the  more  general  expression  for 
any  spherical  versor  is 

$^  =  eosO  -{-  smO\cos <fi{cos ij/ ' p -{■  sini/^-y)  —  sin<^.ap. 

We  observe  that  the  line  OX  is  the  principal  axis  of  the 
product  versor  POQ. 

To  Jind  the  product  of  two  spherical  versor s  of  the  general  form 
given  above. 

The  two  factor  versors  may  be  expressed  by 

IT 

^"  =  cos  u  -\-  sin  u  (cos  <^  •  /?  —  sin  <^  • «)  ^, 

IT 

and  rf  =  cos  v  +  sin  v  (cos  <^'-  y  —  sin  <^'-  a)  ^, 

where  /S  and  y  denote  any  unit  axes  perpendicular  to  a.     The 
product  has  the  form 

^"'=  cosiC'H-  sin  i(;  (cos  <f>"-y—  sin  <^"-  a)  . 

Since  ^"t^''  =  cos  it  cos  v  —  sin  w  sin  v  cos  It; 

—  ff 
+  jcosvsinzt'^-f-coswsin'u-T;  —  sin?t  sini;  sin^>;.4^r;P, 


'■$■>"  Oif 


6      PRINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC  ANALYSIS. 

and  cos  ^rj  =  cos  <^  cos  <^'  cos  (By  -\-  sin  <^  sin  <}>', 

and  Sin  $r}  =  cos  <^  cos  <l>'  sin  /3y  •  /3y 

—  (cos  <^  sin  <f>''l3a-{-  cos  <^'  sin  <^  •  ay), 

therefore     cos  iv  =  cos  u  cos  v 

—  sin  u  sin  v(cos  <^ cos  <^' cos  /8y + sin  <^  sin  <^'),  (1) 

sin  w  sin  <f>"  =  cos  u  sin  v  sin  <^'  -f  cos  ?;  sin  u  sin  <^ 

+  sin  u  sin  2?  cos  <^  cos  <f>'  sin  /8y,  (2) 

sinwcos<^"-€  =  cos w  sin  V cos  c^'-y  ^-cosv sin?^cos<^-/S 

4- sin w sin -y (cos <^  sin <^'- /8a + cos <^' sin <^.  ay).  (3) 

From  equation  (1)  we  obtain  tv,  then  from  (2)  we  obtain  <f>",  and 
finally  from  (3)  we  obtain  c. 

When  the  factor  versors  are  restricted  to  one  plane,  the  axes 
coincide ;  that  is,  r}  =  t     The  above  formula  then  becomes 

^9+0'  _  gQg  ^  cos  0'—  sin  0  sin  6' 

-h  (cos  ^  sin ^'+  cos  &  sin^)  {cos  <^  •  j8  —  sin  <j5)  •  aj  , 

which   is   the   fundamental    theorem    for  trigonometry   in   any 
plane. 

When  the  axes  are  coplanar  with  the  initial  line,  we  have  y 
identical  with  p,  but  <^',  in  general,  different  from  <^.  The  theo- 
rem then  becomes 

^Y  =cos^cos^'-sin^sin^'cos((j!)'-<^) 

+  \  (cos  6  sin^'  cos  <^'+cos^'  sin ^  cos  <^)  •  ^ 

+  sin^sin^'sin(<^'-<^).j8a 

—  (cos  ^  sin  ^' sin  <^' +  cos  ^' sin  ^  cos  <^)  •  a  p. 

If,  in  addition,  the  middle  term  of  the  sine  vanishes,  the  axis 
of  the  product  will  also  be  in  the  same  plane  with  the  other  axes 
and  the  initial  line. 

To  prove  that  the  sum  of  the  squares  of  the  three  components  of 
the  product  of  two  general  spherical  versors  is  unity. 

For  shortness,  let  a;  =  cos  ^,  y  =  sin  0  cos  <^,  z  =  sin  ^  sin  <^ ; 
if'  =  cos  6\   y'  =  sin  0'  cos  <f>',   z'  =  sin  0'  sin  <^'.     Then 


PRINCIPLES  OF   ELLIPTIC  AND   HYPERBOLIC  ANALYSIS.     7 

cos^^"  =  {xx' —  yy'  COS  fty  —  zz'y 

=  x^x'  ^  +  y-y'  ^  cos^ySy  +  zh'  ^  —  2  xx'yy'  cos  ^y  —  2  xx'zz' 
+  2  yy'zz' cos  Py, 
(sill  0"  cos  <!>"•  €y=\xy'  'y  +  x'y'l3-{-  yz'  -^a  —  zy'-ya]^ 

—  a^2/'2  +  a;'y  +  2/"2;'^+2!^2/'"+  ^^x'yy'  cos  (Sy +2  xyy'z'  cos  y^a 

—  2yz'x'y'  cos/8y«  —  2yzy'z'  cos  (3a  •  ya, 
(siii^"siii<^")2  =ja;2;'+x'2;  +  ?/2/'sin^yP 

=  xh''^  +  2;^a;'^  +  2/-?/'^  sin-ySy  +  2  it'a;'2;2;'  +  2  iC2/2/'2!'  sin  )8y 
-\-2x'yy'z  siii/3y. 

The  sum  of  tlie  square  terms  is  (x'^ -\- y'^ -\- z^)  {x'^ -{- y'^ -{- z'^) , 
that  is,  1 ;  and  the  sum  of  the  product  terms  reduces  to 

2yy'zz' (cos fty  —  cos /3a  •  ya)  +  2 xyy'z' (cosy  13a  -\-  sin/8y) 

—  2  yz'x'y' (cos /3ya—  sin^y). 

Now,  p  and  y  both  being  perpendicular  to  a,  cosySy  =  cos /3a  •  ya, 
and  sin^y  =  —  cosy/3ot  =  cos/3ya.  Hence  the  sum  of  the  product 
terms  vanishes. 


FUNDAMENTAL  THEOREM  FOR  THE  ELLIPSOID 
OF  REVOLUTION. 

Imagine  a  circle  APB  (Fig.  5)  to  be  projected  on  the  plane 
of  AQB,  by  means  of  lines  drawn  from  the  points  of  the 
circle,  perpendicular  to  the  plane, 
as  PQ  from  P;  the  projection 
of  the  circle  is  an  ellipse,  hav- 
ing the  initial  line  for  semi-major 
axis.  Let  A.  denote  the  axis  of 
the  circle,  and  ft  that  of  the 
plane  ;  all  lines  perpendicular  to 
the  initial  line  are  in  the  pro- 
jected figure,  diminished  by  the 
ratio  cos\/3,  while  all  lines  parallel 
to  the  initial  line  remain  unal- 
tered. Any  area  A  in  the  circle  will  be  changed  into  A  cos  \/3 
in  the  ellipse ;  and  this  is  true  whatever  the  form  of  the  area. 
For  shortness,  cos  A/3  will  be  denoted  by  k. 


Fig.  5. 


8      PRINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC  ANALYSIS. 

The  projecting  lines,  instead  of  being  drawn  perpendicular  to 
the  plane  of  projection,  may  be  drawn  perpendicular  to  the  plane 
of  the  circle;  the  ratio  of  projection  then  becomes  secAyS,  which 
may  likewise  be  denoted  by  k,  but  k  is  then  always  greater  than 
unity.  The  figure  obtained  is  an  ellipse,  having  the  initial  line 
for  semi-minor  axis.  By  the  revolution  of  the  former  ellipse 
round  the  initial  line  we  obtain  a  prolate  ellipsoid ;  by  the  revo- 
lution of  the  latter,  an  oblate  ellipsoid. 


The  Fundamental  Equation  of  Elliptic  Trigonometry. 
The  elliptic  versor  is  expressed  by  OP  (Fig.  6),  and 

The  problem  is,  to  find  the  correct  analytical  expressions  for 
these  three  terms.     If  by  w  we  denote  the  ratio  of  twice  the  area 

of  the  sector  AOP  to  the  square  on 

OA,  then, 


OM  u 

— —  =  cos- 
OA  k 


and  — —  =  Zcsin 


OA 


k 


Hence,  if  ^  denote  a  unit  axis  nor- 
mal to  the  plane  of  the  ellipse,  the 
equation  may  be  written 


Fig.  6. 


(fc/S)" 


COS-  +  sm-' 
k  k 


{m 


But  we  observe  that  it  is  much  simpler  to  define  u  as  the  ratio  of 
twice  the  area  of  AOP  to  the  rectangle  formed  by  OA  and  OB, 
the  semi-axes  ;  for  then  we  have 

{kp)  "  =  cos  u  -\-Qmu'  {kp)  ^. 

We  attach  the  A;  to  the  axis  rather  than  to  the  ratio,  because  in 
forming  a  product  of  versors  it  does  not  enter  as  an  ordinary 
multiplier.  When  the  elliptic  sector  does  not  start  from  the 
principal  axis,  the  element  u  must  still  be  taken  as  the  ratio  of 
twice  the  area  of  the  sector  to  the  rectangle  formed  by  the  axes. 
The  index  f  is  due  to  the  rectangular  nature  of  the  components ; 
it  expresses  the  circular  versor  between   OA  and  MP.     When 


PRINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC  ANALYSIS.       9 


oblique  components  are  used,  the  index  is  then  w,  the  angle  of 
the  obliquity.     This  is  proved  in  Fundamental  Theorems,  page  10. 

To  find  the  product  of  two  elliptic  versors  which  are  in  one  plane 
passiyig  through  the  principal  axis. 

Let  the  two  versors  be  represented  by  OQA  and  OAP  (Fig.  6); 
then  their  product  is  represented  by  OQP.  Let  p  denote  a  unit 
axis  normal  to  the  plane ;  the  former  versor  may  be  denoted  by 
ikfiy,  and  the  latter  by  {Ivfiy.     Then 

(fe/8)"(A;^)'' =  {cos'M  +  sin?^.(A;/?)^i  Jcosv  +  sin v.(fe/3)^J 

=  cosw  cos-y-f  COS?*  sin?;  •  (A:/8)^4-cos'y  sinw  -(A:/?)^ 

TT  IT 

+  sint*sin?;.(A;y8)^(A;y8)   . 
Now  {kPY{kpY^{kl^Y+' 

=  eos(u  -\-v)  +  sin{u-{-v)'{kfS)^ 
=  cos  ii  cos  V  —  sin  u  sin  v 

TT 

+  (cos  u  sin V  +  cos  v  sin u)  •  {k/3)  ^. 

Hence  {k/BY  (kfSy  =  ft'' =  -  1.  From  this  we  infer  that  A:  is 
such  a  multiplier  that  it  does  not  affect  the  terms  of  the  cosine. 

To  find  the  ^woduct  of  two  elliptic  versors  which  intersect  iii  the 
principal  axis  of  the  ellipsoid  of  revolution. 

Let  — —  OA  and  — —  OQ  (Fig.  7)  represent  the  two  versors ; 
O-t  (JA 

their  axes  are  ft  and  y,  respectively,  each  being  perpendicular  to 
a,  the  direction  of  the  principal 
axis  OA.  Let  u  denote  the  ratio 
of  twice  the  area  of  OPA  to  the 
rectangle  formed  by  the  semi-axes  _r' 
of  its  ellipse,  and  v  the  ratio  of 
twice  the  area  of  OAQ  to  the  rec- 
tangle formed  by  the  semi-axes  of 
its  ellipse.  The  versors  are  denoted 
by  {kftY  and  (fey)".     Now 

(kfty  =  cos  u  +  sin  u  •  (kft)  ^, 

TT 

and  (kyY  —cosv-\-siTiv-{ky)^, 

■n 
therefore  {kftyikyY  =  cosit  cosv  +  cos  v  sinu  •  {kft) 

TT  a:  ^ 

-f  cos?^  sinv  •(A:y)^-f  sin?*  ^\\\v-{kfty{ky). 


10      PRINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC  ANALYSIS. 

By  means  of  the  principle  that  the  first  power  of  k  is  k,  we  see 
that  the  second  and  third  terms  contribute 

k  (cos  V  sin  11-/3  +  cos  usinv-  y) 

to  the  Sine  component.  It  remains  to  determine  the  meaning  of 
the  fourth  term,  that  is,  the  values  of  the  coefficients  x  and  y  in 
the  equation 

{kl3)  ^  (ky)  ^  =  a;  cos  /3y  -h  2/  sin  /3y  •  ySy ^. 

From  the  form  of  the  product  of  two  coplanar  versors  (page  9), 
it  appears  that  x  is  —  1 ;  the  value  of  y  appears  to  be  either  —  k^ 
or  —1. 

On  the  former  hypothesis  the  directed  sine  OB  would  be 

A:cos'y  sinu-  jS  +  kcosu  sinv-y  —  A:^sin^t  sin -u  sin ;8y  a. 

xr 

Now  OF  =  cos  u  -a  —  k  sin  u  -/S^jSy, 

n  

and  OQ  =  cosv-a-j-ksmv-  y^/3y  ; 

consequently     cos  ( OB)  ( OP)  =  -  A:^  cos  v  sin^u  f^^l&L  _  S21&L 

7  9  •  •         /C0S-y8y  1  ,       ■      o 

—  k-  cos  u  sm  w  smv  f  — f^ h  sm  By 

\siir^y      sm/Jy 

which  vanishes,  as  before  (page  3) .  Similarly  cos ( OB)  {0Q)  =  0. 
Hence  the  above  expression  gives  the  direction  of  the  normal  to 

the  plane  of  the  product  versor.  But  suppose  that  —-  OA  and 
-— —  OQ  are  quadrantal  elliptic  versors,  then  gosu  =  cos  v  =  0,  and 
sin  u  =  sin  v  =  1 ;  consequently  the  cosine  of  the  product  would 

then  be  —  cos  ^y  and  the  sine  of  the  product  —  k^  sin^y  •  a  .  But 
it  is  evident  that  in  this  case  the  product  versor  is  circular, 

namely,  —  (cos/8y  +  sinySy  •  a^).     Hence  it  appears  that  F  cannot 
enter  as  a  factor  of  the  third  term  of  the  Sine. 
On  the  other  hypothesis  the  directed  sine  is 

k{cosvsmU'  13  +  coswsini;-y)  —  sin  %  sin  v  sin^ya. 

This  expression  satisfies  the  test  of  becoming  circular  under 
the  conditions  mentioned ;  but  its  direction  is  not  normal  to  the 


PRINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC  ANALYSIS.      11 

plane  of  the  product  versor.  How  then,  is  its  direction  related 
to  that  plane  ?  It  will  be  found  that  it  has  the  direction  of  the 
conjugate  axis  to  the  plane.  Draw  OV  (Fig.  8),  to  represent 
k (cos  V  sin  u  •  /?  +  cos  usinv-y),  the  component  perpendicular  to 
the  principal  axis  OA,  and  OU'  in  the  direction  opposite  to  the 


principal  axis  to  represent  —  sinw  sinv  smjSy,  also  OU"  to  repre- 
sent the  same  quantity  multiplied  by  k^ ;  and  draw  OR'  and  OR, 
the  two  resultants.  The  plane  through  OA  and  OV  will  cut 
the  ellipsoid  in  a  principal  ellipse  AXB,  and  as  it  passes  through 
the  normal  OR  it  will  cut  the  plane  of  the  product  ellipse  at 
right  angles ;  let  OX  denote  the  line  of  intersection.  Draw  XA' 
parallel  to  OA  and  XD  the  tangent  at  X,  and  let  0  denote  the 
circular  versor  between  AO  and  OX.     Now 


tan^  =  ^^=^ 
OM     OV 


k  sinu  sinv  sin^y 


Vcos^v  sm^u  +  cos^i*  sin^v  +  2  cos  u  cosv  sin  u  sinv  cos  /3y 
but  tan  A'XD  =  -k^  cotan  $ 

_  _  k^GOs^v  sin^it-f  cosV  sin^^  +  2  cos  u  cost;  sinu  sin'?;  cos /?y 
siiiu  sin'y  sin^y 

=  cotan  VOR'  =  tan  ^Oi^'. 

Thus  the  direction  of  OR'  is  that  of  the  conjugate  axis  of  the 
plane  of  the  product  versor. 

Let  <l>  denote  the  ratio  of  twice  the  area  of  AOX  to  the  square 
of  OA ;  it  is  equal  to  the  angle  which  OX  made  with  OA  before 
the  contraction.  The  direction  of  the  axis  was  then  cos  <^  along 
OB,  and  sin<^  along  OA' ;   by  the   contraction,  cos<;^   has  been 


12      PRINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC  ANALYSIS. 

changed  into  k  cos  <^ ;  hence  the  axis  of  the  ellipsoid,  along  the 
direction  of  OR',  is  A;  cos  <^  •  c  —  sin  <^  •  a,  where  e  denotes  a  unit 
axis  in  the  direction  of  OB. 

The  magnitude  of  the  product  versor  is  determined  by  the 
cosine  function, 

cosw  cos-y  —  smu  sinv  cosjSy. 

Suppose  that  an  elliptic  sector  OXZ  (Fig.  7),  having  the  area  of 
the  third  side  of  the  ellipsoidal  triangle,  starts  from  the  semi- 
major  axis  OX,  and  let  OY  and  OZ  be  the  rectangular  projec- 
tions of  the  bounding  radius  vector  OZ.  As  the  small  ellipse 
OPQ  is  derived  from  a  principal  ellipse  by  diminishing  all  lines 
parallel  to  OX  in  the  ratio  of  OX  to  OA,  that  is,  in  the  ratio  of 
Vcos^<^  +  k-  sin^^  to  1,  while  the  transverse  lines  remain  unal- 
tered; the  ratio  of  OF  to  OX  is  equal  to  the  corresponding  ratio 
in  the  principal  ellipse ;  hence  the  ratio  of  OY  to  OX  is  equal  to 
cos  u  cos  V  —  sin  u  sin  v  cos  jSy. 

Let  tv  denote  the  ratio  of  twice  the  sector  OPQ  to  the  rectangle 
formed  by  OX  and  the  minor  semi-axis  of  the  ellipse  OPQ ;  this 
ratio  is  equal  to  the  ratio  of  twice  the  corresponding  circular  sec- 
tor to  the  square  of  OA.  By  the  corresponding  circular  sector  is 
meant  that  circular  sector  from  which  the  elliptic  sector  was 
formed  by  contraction  along  the  two  axes.  Also,  let  $  denote 
the  elliptic  axis,  cos  cfj  >  ke  —  sin  <f)  -  a.  The  product  versor  then 
takes  the  form 

n 

^"^  =  GOSw  +sinw(cos<^-A:£  —  sin<^««)^, 
the  quantities  iv,  (f>,  and  e  being  determined  by 

cos  ic  =  cos  u  cos  V  —  sin  u  sin  v  cos  )8y,  (1 ) 

sm.»  =  ^'""^^°^"^"^y  (2) 

Vl  —  cos^w 

—  CQS'^sini^-^  +  cos^  sini;'Y  .o\ 

sin  w  cos  <^ 

Consequently  we  have  for  the  elliptic  axis  OP, 

^_k{Gosv  sinu- (^-{-  cosi^sin^-y)  —  sinu  sinv  sin fSy  a 
Vl  —  cosho 


PRINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC  ANALYSIS.      13 

The  locus  of  the  poles  of  the  several  elliptic  areas  is  the  original 
ellipsoid. 

To  find  the  product  of  two  ellipsoidal  versors  of  the  above  general 
form. 

The  two  factor  versors  are  expressed  by 

n 

^«  =  cos?^+  sinw(cos<^-A;y8—  sin<^-«)^, 
and         yf  =  cos  v  +  sin  v  (cos  cf>'  -ky  —  sin  <^'  • «)  ^  ; 
it  is  required  to  show  that  their  product  has  the  form 

rr 

^"  =  cos  10  +  sinw(cos(^"-A:e  — sin(^"-a)  ^. 
We  have 

i"7f  =  (gosu  +  sin?/-^^)  (cosv  +  sinv-r;^) 
=  cosw  cosv  —  sini^  sin  v  cos^ry 

n 

+  \cosu  sin -u- 77  +  cos'U  sinit-^  —  sin  it  sin  1;  Sin  ^7;  p. 

The  problem  is  reduced  to  finding  the  value  of  cos  ^7;  and 
Sin  ^7;.  Now  $rj  means  the  elliptic  versor  between  the  elliptic 
axes 

cos (ft-k/^  —  sin <^ •  a     and     cos cf>'  -ky  —  sin <f>' •  a. 

To  find  them,  we  apply  the  following  principle : 

Restore  the  elliptic  axes  to  their  spherical  originals,  find  the 
versor  between  these  unit  axes  according  to  the  ordinary  rule, 
and  reduce  its  axes  back  to  the  ellipsoidal  form.  Applied  to  the 
above,  the  rule  means :  suppose  k  =  1,  form  the  cosine  and  the 
directed  Sine,  and  introduce  k  as  a  multiplier  of  those  components 
of  the  directed  Sine  which  are  perpendicular  to  a.     Hence 

cos  $ri  =  cos  <^  cos  <^'  cos  I3y  +  sin  ^  sin  <^', 
and      Sin  $r]  =  cos  <^  cos  <l>'  sin  fty  •  a 

—  k(cos(fiSm<f>' '  fta-\-sincf}GOS(f>'  -ay). 

If  we  express  Sin  It;  as  sin^Ty^/;,  what  must  ^77  now  mean? 
Its  length  is  not  unity,  nor  is  it  normal  to  the  plane  of  |  and  7;. 
It  means 

cos  <^  cos  <^^  sin/?y«  —  k(cos<f>  sm<f)'  •  (3a  +  sin<^  cos<^^- oty), 

that  is,  the  elliptic  axis  conjugate  to  the  plane  of  $  and  -q. 


14      PRINCIPLES  OF  ELLIPTIC  AND  HYPEKBOLIC  ANALYSIS. 

Hence 
cosw=cos?^cos?;— sini^  sinv(cos<^  cos<^'cos/3y  +  sin<^  sin<^'),  (1) 

sin  w  sin  <^"     =  cos  u  sin  v  sin  <^'  +  cos  v  sin  u  sin  <^ 

+  sin  u  sin  v  cos  <^  cos  <^'  sin  fBy,  (2) 

sin  w  sin  <j)"  'c  =  cos  i^  sin-y  cos  <^'  •  y  +  cos  v  sin  u  cos  </>  •  (i 

+  sinwsinv(cos<^sin<^'')8a  +  cos<j!>'sin(^«ay).     (3) 

FUNDAMENTAL  THEOREM  FOR  THE  GENERAL 
ELLIPSOID. 

To  find  the  product  of  tivo  ellipsoidal  versors  vjhose  axes  have  the 
same  directions  as  the  minor  axes  of  the  ellipsoid. 

In  the  general  ellipsoid  there  are  three  principal  axes  mutually 
rectangular ;  in  Fig.  9  they  are  represented  by  OA,  OB,  00.  We 
shall  suppose  the  greatest  semi-axis  to  be  taken  as  the  initial  line, 
but  either  of  the  others  might  be  chosen. 
Let  unit  axes  along  OA,  OB,  and  00  be 
denoted  by  a,  (B,  y,  respectively ;  let  k^  de- 
note the  ratio  of  OB  to  OA,  and  k  that  of 
00  to  OA.  A  versor  POA  in  the  plane 
CO  A  is  expressed  by  {k^Y,  while  a  versor 
AOQ  in  the  plane  of  ^0-B  is  expressed  by 
(Zj'y)";  u  denoting  the  ratio  of  twice  POA 
to  the  rectangle  CO  A,  and  v  that  of  twice  AOQ  to  the  rectangle 
AOB. 

Now  (A:^)"(A:'y)'=Scosw  +  sinw.(ifcy8)^Hcos'y-f  sinv-(A;'y)|^ 
=  COS w COS V  -f  cos-v  sinu' (k^)^ 

+  COS  u  sin  V  •  {k'y)  ^  -f  sin  u  sin^?  •  (k^)  ^  (k'y)  ^. 

The  fourth  term,  as  it  involves  two  axes  which  are  at  right 
angles,  can  contribute  nothing  to  the  cosine  ;  the  cosine  is 
cos w  cosv.  The  second  and  third  terms  contribute  kcosv  sinu- /3 
-f  k'  cos usinv-y  to  the  directed  Sine ;  while  the  fourth  con- 
tributes either  —  kk'  sinu  sin-y  •  a  or  —  sin?^  sin v  •  a. 

It  may  be  shown,  in  the  same  manner  as  before  (page  2),  that 

k  cos  V  sin  u- 13  +  k'  cos  it  sin  'y  •  y  —  kk'  sin  u  sin  v  •  a 


Fig.  9. 


PRINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC  ANALYSIS.    15 

is  perpendicular  to  both  OP  and  OQ,  hence  has  the  direction  of 
the  normal  to  their  plane ;  and,  by  the  principle  stated  at  page 
13,  it  is  seen  that 

fccosv  sini6-/8  + fc'cosi^  sin-y-y  —  Sinwsinv.« 

is  the  axis  conjugate  to  the  plane  of  POQ. 

Let  a  plane  pass  through  the  principal  axis  and  the  perpen- 
dicular component  k  cos  v  sin  U'  f3  -\-  k'  cos  u  sin  v-y;  as  it  passes 
through  the  normal  to  the  plane  POQ  it  must  cut  that  plane  at 
right  angles,  and  OX,  the  line  of  intersection,  is  the  principal 
axis  of  the  ellipse  PQ.  Let  <^  denote  the  elliptic  ratio  of  AOX, 
and  ij/  the  angle  between  jS  and  cos  v  sin  u-  (3  -{■  cos  u  sin  v  •  y,  and  w 
the  ratio  of  twice  the  elliptic  versor  POQ  to  the  rectangle  of  the 
semi-axes  of  its  ellipse;  then  the  product  versor  takes  the  form 

$'"  =  cosw  +  sinz(;Jcos<^(A;cosj//-/5  +  A;'sini/^- y)  —  sin<^  •  «J  ^. 

For  cos  10  =  cos  w  cos  v,  (1) 

sin  w  sin  <f>  =  sin  u  sin  v,  (2) 

sin  w  cos  <^  cos  j/a  =  cos  v  sin  u,  (3) 

sinwcos<^  sin»/^  =  cosi^  sinv.  (4) 

To  find  the  product  of  two  ellipsoidal  versors  of  the  above  form. 
Let  the  one  versor  be  ^",  where 

$  =  cos  <^  (A;  cos  ij/ '  ^  -]-  k'  sinij/ '  y)  —  sin  cf>  •  a, 
and  let  the  other  be  ry'',  where 

77  =  cos<^'(A:cosj/''-y8-}- A;' sin»//'-y)  — sin<^'- «;    . 
it  is  required  to  show  that  ^V  has  the  form  C%  where 

^  =  cos  0"(A:  cos ij/" '  (3  -{-  k'  sin \p  -y)  —  sin <^"  •  a. 

Since  ^""rf  =  cos  u  cos  v  —  sin  u  sin  v  cos  ^-q 

-f-  J  cos  'y  sin  ?^  •  ^  -f  cos  1^  sin  'y .  r;  —  sin  u  sin  v  Sin  ^7;  J  ^, 

the  problem  reduces  to  finding  cos  ^77  and  Sin^i;.  By  ^rj  is  meant 
the  elliptic  angle  between  the  elliptic  axes  ^  and  -q ;  the  ratio  of 
the  sector  ^-q  to  the  rectangle  of  its  ellipse  is  the  same  as  the 
ratio  of  the  sector  of  the  primitives  of  i  and  -q  to  1.  Hence  the 
cosine  is  obtained  by  supposing  k  and  A:'  to  be  one,  and  the  Sine  is 


16      PRINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC  ANALYSIS. 

obtained  by  the  same  method,  and  then  reducing  by  k  the  compo- 
nent having  the  axis  /?,  and  by  k'  the  component  having  the 
axis  y.     We  obtain 

cos  ^7)  =  cos  <^  cos  <^'  cos  {xp  —  i//')  +  sin  <^  sin  <f>\ 

and  Sin  ^-q  =  cos  <^  cos  <^'  sin  (i//  —  i//' )  • « 

-f  k\(io^<ji  cosj/Asin<^'— cos<^'cosj/^'sin<^)-y 

—  k  (cos  (^  sin  \p  sin  <^'  —  cos  <^'  sin  i//'  sin  <^)  •  /?. 
Hence  cos  w 

=  cos  w  COS  V  —  sin  i^  sin  v  J  cos  <^  cos  <^'  cos  (i//  —  »//')  +  sin  <^  sin  cf>'\,(l) 

sinK;cos<^"cosj/A" 

=  cos  u  sin  V  cos  <^'  cos  i//'  -\-  cos  v  sin z*  cos  <^  cos  \}/ 

-f  sini*sinv(cos<^sini/^sin<^'  — cos<^'sinj/A'sin<^)j  (2) 

siniycos<^"sinj//" 

=  cos  u  sin  V  cos  <^'  sin  ij/'  +  cos  v  sin  u  cos  <^  sin  i(/ 

—  sin  I*  sin  v  (cos  <j!)  cos  «/^  sin  </>'  —  cos  <^'  cos  i/a'  sin  </>) ,  (3) 

sin2(;sin<^"  =  cosw  sin-u  sin</)'  +  cosv  sin  w  sin  <^ 

—  sin  u  sin  v  cos  <^  cos  <^'  sin  (i//  —  i//') .  (4) 

The   elliptic   axis   is   given   in   magnitude   and    direction    by 
^^^-^^2 The  locus  of  these  axes  is  an  ellipsoid  derived 


Vl  —  COS^^ry 

from  the  original  ellipsoid  by  interchanging  the  ratios  k  and  k\ 


FUNDAMENTAL    THEOREM    FOR   THE    EQUILAT- 
ERAL HYPERBOLOID  OF  T^WO  SHEETS. 

In  order  to  distinguish  readily  the  equilateral  from  the  general 
hyperbola,  it  is  desirable  to  have  a  single  term  for  the  equilateral 
hyperbola.  The  term  exdrcle,  with  the  corresponding  adjective 
excircular,  have  been  introduced  by  Mr.  Hay  ward,  in  his  "Algebra 
of  Coplanar  Vectors."  These  terms  are  brief  and  suggestive,  for 
the  equilateral  hyperbola  is  the  analogue  of  the  circle.  If  we 
consider  the  sphere,  we  find  that  its  hyperbolic  analogue  consists 
of  three  sheets.  Two  of  these  are  similar,  the  one  being  merely 
the  negative  of  the  other  with  respect  to  the  centre,  and  are 
classed  together  as  the  equilateral  hyperboloid  of  two  sheets ;  the 


PRINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC  ANALYSIS.      17 


third  is  called  the  equilateral  hyperboloid  of  one  sheet.  For 
brevity  we  propose  to  call  these  the  exsphere  of  two  sheets,  and  the 
exsphere  of  07ie  sheet,  the  two  together  being  called  the  exsphere. 
In  treating  of  the  exsphere  of  two  sheets,  we  shall  generally 
consider  the  positive  sheet. 

To  find  the  expression  for  an  exspherical  versor,  the  plane  of  which 
passes  through  the  principal  axis. 

Let  OA  (Fig.  10)  be  the  principal  axis  of  an  equilateral  hyper- 
boloid of  two  sheets,  QAP  a  section  through  OA,  AOP  the  sector 
of  a  versor  in  that  plane,  and  PM 
perpendicular  to  OA.    The  versor  is 

denoted  by  -^  OP,  or  (OA){OP), 


OA 

if  OA  is  of  unit  length. 


Now 


-^  0P=  -^  (OJf  +  MP) 
OA  OA^        ^        ^ 

OA  ^  OA 

The  problem  is  to  find  the  proper 
analytical  expression  for  this  equa- 
tion. Let  ^  denote  a  unit  axis 
normal  to  the  plane  of  QAP,  and 
u  the  ratio  of  twice  the  area  of  the 
sector  AOP  to  the  square  of  OA, 
or  rather  to  the  area  of  the  rec- 
tangle AOB,  and  let  i  denote  V  — 1. 
starting  line  is  indifferent,  is  expressed  by 

^'"  =  cos  m  +  sm  m  .  y8^ 

=  cosh  u  +  i  sinh  u-  ^^. 

OM  •  MP  f 

We  observe   that    coshu  =  -—-,  and  smhu  =  ——,  and  that  /5 

expresses  the  circular  versor  between  OA  and  MP.  What  is  the 
geometrical  meaning  of  the  i?  It  expresses  the  fact  that  coshw 
and  sinh?*  are  related,  not  by  the  condition 

cosh'^w  4-  ^mh^u  =  1, 

but  by  the  condition  cosh^w  —  sinh^w  =  1. 


The  above  equation,  if  the 


18      PRINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC  ANALYSIS. 

With  this  notation,  we  can  deduce  readily  from  any  spherical 
theorem  the  corresponding  exspherical  theorem. 

A  plausible  hypothesis  is  that  the  i  before  sinh  u  may  be  con- 
sidered as  an  index  f  to  be  given  to  the  axis  (i,  making 

y8'"  =  cosh  u  -\-  sinh  w  •  ^'^ ; 

but  this  would  leave  out  entirely  the  axis  of  the  plane,  for  the 
equation  would  reduce  to 

yS*"  =  cosh  it  —  sinhw. 

The  quantity  here  denoted  by  i  is  the  scalar  V  — 1,  while  the 
index  f  expresses  the  vector  V  — 1. 

The  series  for  e'"  is  wholly  scalar;  but  the  series  for  e'"-^^ 
breaks  up  into  a  scalar  and  a  vector  part. 

In  specifying  an  exspherical  versor,  it  is  necessary  to  give  not 
only  the  ratio  and  the  perpendicular  axis  of  the  plane,  but  also 
the  principal  axis  of  the  versor.  This  is  the  reason  why  the 
spherical  versor  has  to  be  treated  with  reference  to  a  principal 
axis,  in  order  to  obtain  theorems  which  can  be  translated  into 
theorems  for  the  exspherical  versor. 

To  find  the  product  of  two  coplanar  exspherical  versor s,  when  the 
common  plane  passes  through  the  principal  axis. 

Suppose  the  versors  shifted  without  change  of  area  until  the 
line  of  meeting  coincides  with  the  principal  axis.  Let  QOA 
(Fig.  10)  be  denoted  by  /?"',  and  ^ OP  by  yS""",  expressions  which 
are  independent  of  the  shifting.     Then 

K 

^■'*  =  coshw  +  isinhu'  /3'^, 
^"^  =  cosh  V  +  i  sinh  v  -  (3   ; 
therefore     ^8*"^'"  =  (cosh  ic  +  i  sinh  u-  p^)  (cosh  v  +  sinh  v- 13  ) 

n 

=  cosh?^cosh'V^-^(coshwsinhv^-cosh^'sinh^^)•^^ 
-f  i^  sinh  u  sinh  v  -  J3''  \ 

but  i^  =  — ,  and  /J""  =  —  ;  hence 

piuj^iv  _  (,Qg]^  ^^  cosh  V  +  sinhw  sinh  v 

IT 

+  i{q,o^\\u  sinhv  -f-  coshr  sinhw)-/8^. 
Hence       ^"^'■^  =  ^«(«+''). 


PKINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC  ANALYSIS.      19 

Suppose  that  the  sector  QOP  is  shifted  without  change  of  area 
till  it  starts  from  OA,  and  becomes  AOR.     Then 

=  cosh  u  cosh  V  4-  sinh  u  sinh  v, 

OA  ' 

and  =  cosh  u  sinh  v  -j-  cosh  v  sinh  u. 

OA 

To  find  the  product  of  two  diplanar  exspherical  versors  when  the 
X>lane  of  each  passes  through  the  principal  axis. 

Let  the  two  versors  POA  and  AOQ  (Fig.  11)  be  denoted  by  /J"" 
and  y%  the  axes  f3  and  y  being  each  perpendicular  to  the  princi- 
pal axis  a.     Then 

yg^y  =  (cos  iu  +  sin  iu-  (S^)  (cos  iv  +  sin  iv-y^) 

=  cosm  cosiv  —  sinm  sin  w  cos^y 

+  \  cos  iv  sin  m  •  ;8 + cos  m  sin  iv  •  y — sin  iu  sin  I'v  sin^y  •  «  ^ . 
But  cos  iu  =  coshii,  and  sin  iu  —  i  sinh  w,  therefore, 
^uyv  _  cosh  It  cosh'U  +  sinhw  sinhv  cos^y, 

TT 

+ 1 J  cosh  v  sinh  It  •  /?  +  coshw  sinhv  ^y— i  sinh  it  sinhv  sin^Sy  •  ctp. 
Hence  coshjS'^y'"  =  cosh?^  coshv  +  sinhw  sinli-u  cos/3y 

and       Sinh  jS^'^y'"  =  cosh  v  sinh  u-fS  -{-  cosh  u  sinh v  •  y 

—  i  sinh  ii  sinh  v  sin  ^y  a. 

By  expanding,  it  may  be  shown  that 

(cosh^"y-)2-  (Sinh/3V)'  =  1^ 

or  (cos^'VO^  +(Sin^*V'')^    =  1- 

The  function  Sinh  is  the  same  as  Sin,  only  an  i  has  been 
dropped  from  all  the  terms  of  the  latter.  The  product  versor 
is  also  represented  by  a  sector  of  an  excircle  of  unit  semi-axis. 

The  first  and  second  components  of  the  excircular  Sine  are  per- 
pendicular to  the  principal  axis ;  hence  their  resultant, 

cosh  V  sinh  w  •  /8  +  cosh  u  sinh  v  •  y, 

is  also  perpendicular  to  the  principal  axis.  Let  it  be  represented 
by  OF  (Fig.  11).  The  diificulty  consists  in  finding  the  true 
direction  of  the  third  component,  —  isinhw  sinh-u  sin^Sy  a.     At 


20      PRINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC  ANALYSIS. 

page  53  of  The  Imaginary  of  Algebra,  I  suggested  the  following 
construction : 

With  V  as  centre,  and  radius  equal  to  sinh  w  sinh  v  sin  ySy, 
describe  a  circle  in  the  plane  of  OA  and  OV,  and  draw  OS  Or  OS^ 
a  tangent  to  this  circle. 

But  another  hypothesis  presents  itself;  namely,  to  make  the 
same  construction  as  in  the  case  of  the  sphere. 


Draw  OU"  opposite  to  OA,  and  equal  to  sinh  w  sinh  ?;  sin  ^y ; 
and  find  OR,  the  resultant  of  OF  and  OU.  We  shall  show  that 
OR  satisfies  the  condition  of  being  normal  to  the  plane  POQ, 
while  OS  or  OS^  does  not. 

The  reasoning  at  page  2  applies  to  give  the  expression  for 
the  vectors  OP  and  OQ.  Hence  the  expressions  for  the  three 
vectors  Oi2,  OP,  OQ,  are 

OR  =  cosh V  sinh u- ^  -{-  cosh u  sinh v-y  —  sinh u sinh v  sin /3y  •  fty, 

0P  =  -  sinhu^-^^'B  +  sinhw  -;-i— •  y  +  cosht^  .^, 
sm  Py  sm  I3y 


OQ  =  — sinhv 


3  —  sinh  v  ^^^-^  •  y  +  cosh  v  •  By. 

sin^y   ^  sin^y    ^  ^^ 


It  follows,  as  there,  that 

cos{OR)(OP)  =  0,     and     cos  (OR)  (OQ)  =  0. 

Hence  OR  is  normal  to  the  plane  POQ,  and  OS  is  not. 

The  function  of  the  i  before  the  third  component  of  the  Sine 
is  to  indicate  that  the  magnitude  of  the  Sine  is  not  -\/0V^  +  VR^ 
but  -VOV'-  VRK     This  gives 


PRINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC  ANALYSIS.      21 


sinh^^Y 


/{cosh^v  sinh^w  +  cosh^w  sinh^-y  +  2  cosh u  coshv  sinh u  sinhtj  cos/37 

—  sinh^M  sinh2i>  sin2/37} 


=  \/(coshwcosh'y  +  sinh  m  sinhv  cos  ^37)^ 
OR 


The  expression 


gives  the  excircular  axis  both 


in  magnitude  and  direction.  The  plane  of  OA  and  OV  cuts  the 
exsphere  in  an  excircle,  and  as  it  passes  through  the  normal  OR, 
it  must  cut  the  plane  POQ  at  right  angles.  Let  OX  be  the  line 
of  intersection    (Fig.    12).      Draw  XM  perpendicular  to    OA-^ 


draw  XD  a  tangent  to  the  excircle  at  X,  and  XA'  parallel  to 
OA,  and  OR'  the  reflection  of  OR  with  respect  to  OV.  Let  <^ 
denote  the  excircular  angle  of  AOX;  that  is,  the  ratio  of  twice 
the  area  of  AOX  to  the  square  of  OA. 

As  OR  is  normal  to  the  plane  POQ,  it  is  perpendicular  to  OX; 
but  OF  is  perpendicular  to  OA;  therefore  the  angle  AOX  is 
equal  to  the  angle  VOR.  Also  as  the  angle  AOR'  is  the  com- 
plement of  i^'OF and  A'XD  the  complement  of  AOX,  the  line 
OR'  is  parallel  to  the  tangent  XD. 

OV 


Hence  cosh  (h  —  —^  -. 
OA 


VOV'-VR' 


4 


and 


cosh^i?  sinh%-|-cosh-?^  sinh^'U  +  2coshi^  cosh i;  sinh u  sinh?;  cos/8y 
(coshw  coshv  +  sinhw  sinhv  eosfSyy  —  1 

MX  VR 


sinh  <^  = 


OA      ^OV'-VR' 


sinh  u  sinh  v  sin  (3y 


V  (cosh  u  cosh  V  +  sinh  u  sinh  v  cos  /3y )  ^  —  1 


(f^^ir  Tar 


22      PRINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC  ANALYSIS. 

The  above  analysis  shows  that  the  product  versor  of  POQ 
may  be  specified  by  three  elements :  first,  c  a  unit  axis  drawn 
perpendicular  to  OA  in  the  plane  of  OA  and  the  normal  to  the 
plane  of  POQ',  second,  <f)  the  excircular  angle  of  AOX  determined 
by  OA  and  OX  drawn  at  right  angles  to  the  normal  in  the  plane 
of  OA  and  the  normal ;  third,  w  the  versor  of  a  unit  excircle 
determined  by  the  conditions  of  passing  through  the  points  P 
and  Q  and  having  its  vertex  on  the  line  OX. 

When  u  and  v  are  equal,  half  of  the  line  joining  PQ  is  the 
sinh  of  half  of  the  versor  of  the  product.  Let  y  denote  the  sinh 
of  each  of  the  factor  versors,  then  it  is  easy  to  see  from  geomet- 
rical considerations  (v.  The  Imaginary  of  Algebra,  page  53),  that 

w 


sinh-  =  —  2/  Vl  +  cos  fSy 


therefore  cosh^  =  — V2 +  2/'(l +cos/^y) ' 

But  it  is  also  evident  that  the  distance  from  O  to  the  mid- 
point of  PQ  is 


4 


/(l-COS/3y)  +  2(/  +  l) 
^2(l  +  C0S/8y)-f2 


The  excess  of  this  distance  over  cosh—  gives  the  distance  by 

z 
which  the  axis  has  been  displaced  along  OX. 

Hence  the  product  versor  may  be  expressed  by  an  excircular 
axis  and  an  excircular  versor  as  i''",  Avhere 

$  =  cosh  <f>-€  —  i  sinh  cfya. 

To  determine  these  quantities,  we  have,  as  in  the  case  of  the 
sphere,  the  three  equations 

cosh  w  =  cosh  u  cosh  v  -{-  sinh  u  sinh  v  cos  f3y,      ( 1 ) 

sinhw  cosh<^  =  sinlut  sinh  v  sinfty,  (2) 

sinh  w  sinh  <^  •  e  =  cosh  v  sinh  w  •  y8  +  cosh  u  sinh  v  •  y.        (3) 

The  axis  e  may  be  expressed  in  terms  of  two  axes  ft  and  y 
forming  with  a  a  set  of  mutually  rectangular  axes,  and  the  angle 
if/  which  it  makes  with  jS',  so  that  for  the  excircular  axis  we 
have 

^=cosh<^(cosj/A.^H-sini/^.y)  —  ?'sinh<^-«. 


PRINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC   ANALYSIS.      23 

In  the  above  investigation  it  is  assumed  that  the  magnitude 
of  the  perpendicular  component  of  the  Sine  is  necessarily  greater 
than  the  component  parallel  to  the  principal  axis.  This  means 
that 

cosIi^t;  sinh^w  +  cosh^w  sinh-v  +  2  cosh  u  cosh  v  sinh  u  sinh  v  cos  /8y 
is  necessarily  greater  than  sinh^i^  sinh^v  sin^/Jy. 

Let  sinySy  =  1 ;  then  cos/3y  =  0  ;  and  we  have  to  compare 
cosh^'y  sinh^w  +  cosh^w  sinh^v    with   sinhV  sinh^v. 

Now  each  term  on  the  left  is  greater  than  the  term  on  the  right ; 
therefore  their  sum  must  be  greater,  for  each  term  is  the  square  of 
a  real  quantity.  Next  let  sin^y  =  0;  then  cos/?y  =  l;  the  for- 
mer term  becomes  a  complete  square  while  the  latter  is  0 ;  hence 
the  former  must  always  be  greater  than  the  latter. 

To  find  the  product  of  two  exspherical  versors  of  the  general  kind. 
The  two  versors  are  expressed  by 

K. 

^"*  =  cosh u-\-i  sinh u (cosh (ft-  fS  —  i  sinh (jy-a)^, 

IT 

and  7)'"  =  cosh v  -]-i  sinh v (cosh (f>' -y  —  i sinh (f>' - a)^  ; 

it  is  required  to  show  that  their  product  has  the  form 

^'"'  =:  cosh  w  -h  i  sinh  to  (cosh  <f>"  -e  —  i  sinh  <^"  • «)  ^. 

We  have  ^'"  =  coshtt  +  i  sinhu  -^^ 

and  7)'"  =  cosh  v-{-i  sinh  v-r}^, 

therefore 

^U^iV        _       (,Qg]^    ^      (,Qg]^    ^       _J_       g^J^]^    ^      gj  J^]^    ^       gQg     ^^ 

TT 

-f  i  \  cosh  u  sinh  V'r)-{- cosh  v  sinh  U'i—i  sinh  u  sinh  v  sin  $rj  •  ^r;  j   • 
It  remains  to  determine  cos  irj  and  Sin  ^rj. 
Since         ^  =  cosh  <f)-  (3  —  i  sinh  <^  •  a, 
and  r;  =  cosh <f>'  -y  —  i  sinh <^'  •  a, 

and  as  we  have  seen  that  the  i  is  merely  scalar,  and  does  not 
affect  the  direction,  we  conclude  that 


24      PRINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC  ANALYSIS. 

COS  $ri  =  cosh  <^  cosh  <^'  cos  (3y  —  sinh  <^  sinh  (f>', 

Singly  =  cosh<^  cosh<^'  sin^y  •  a 

—  I  (cosh  <^  sinh  <^' •  |S«  + cosh  <^' sinh  <^ -ay). 

Substituting  these  values  of  cos  irj  and  Sin  Iry,  we  obtain 

coshz<;  =  coshi^  cosh-u 

+  sinh  u  sinh -y  (cosh  <^  cosh  <^' cos  ;8y— sinh  <j«)  sinh  <^'),   (1) 

sinhwj  sinh<^"  =  cosh?^  sinhv  sinh<^'+  cosh-u  sinh  w  sinh  <^ 

4-  sinh  u  sinh  v  cosh  <^  cosh  <^ '  sin  jSy,  (2) 

sinh  to  cosh<^"-  e  =  cosh  w  sinh  v  cosh  <^'«  y + cosh  v  sinh  w  cosh<^  •  fS 

—  sinh w  sinh V (cosh <^  sinh  <^'- j8a4- cosh <^' sinh </)•  ay). (3) 

Let  us  consider,  more  minutely,  the  above  equations 

cos  $rj  =  cosh  <^  cosh  <^'  cos  /8y  —  sinh  <^  sinh  cfi', 

and    Singly  =  cosh <^ cosh <^' sin /8y.  a 

—  i  (cosh  <^  sinh  4>''f3a-\-  cosh  <^'  sinh  <^  •  ay) . 

If  we  square  these  functions,  we  find 

(cos  ^77)^  =  cosh^c^  cosh^<^' cos^^y  +  sinh^<^  sinh2<^' 

—  2  cosh  <^  cosh  (^' sinh  <^  sinh  </)' cos  ySy, 

(Sin^>;)2  =  cosh^c^  cosh^c^'  sin^ySy  —  cosh^  sinh^c^'  —  cosh^<^'  sinh^<^ 

—  2  cosh  <^  cosh  <f>'  sinh  <f>  sinh  <^'  cos  (ia  ay  ; 

but  cos /8a  ay  =  —  cos/8y,  and  cosh^  =  1  +  sinh^,  therefore, 

As  the  symbol  i  does  not  affect  the  geometrical  composition, 
Sin  ^7;  must  be  normal  to  the  plane  of  $  and  rj;  hence,  if  we 
analyze  it  into  sin  It; -^r;,  we  must  have  sin|7;  =  Vl—  (cosl^?;)^, 
and^- Sinl.; 


V1-(C0S|t;)^ 

Consider  the  special  case,  when  y  =  (3.     Then 
Gosir)  =  CO sh<^  cosh  <;/)'—  sinh  <^  sinh  <^', 
and  Sin^T;  =  —  i(cosh  <f>  sinh  <^'  —  cosh  (f>'  sinh  <^)  f3a. 


PRINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC  ANALYSIS. 


25 


Hence  It;  becomes  an  excircular  versor.    Consider  next  the  special 
case  where  y  is  perpendicular  to  yS.     Then 

cos  ^77  =  ~  sinh  <^  sinh  <f>', 

and  Sin^r;  =  cosh  <^ cosh <^'- aH-t(cosh<^ sinh <^'-y4-cosh<^'sinh<^.y8). 

It  appears  that  the  locus  of  the  poles  of  all  the  axes  is  the 
equilateral  hyperboloid  of  one  sheet,     (v.  page  27.) 


FUNDAMENTAL  THEOREM  FOR  THE  EQUILAT- 
ERAL HYPERBOLOID  OF  ONE  SHEET. 

To  find  the  product  of  a  circular  and  an  excircular  versor,  when 
they  have  a  common  plane. 


K         MLA 
Fia.  13. 

Let  AOP  represent  a  circular,  and  POQ  an  excircular,  versor 
(Fig.  13) ;  and  let  them  be  denoted  by  (S^  and  /5'^     We  have 

l^upv  ^  i^ui-iv  ^  (cos  w  +  sin  w .  (3^)  (cosh'y  +  i  sinh  v  •  fS^) 
=  cos  u  cosh  V  —  i  sin  u  sinh  v 

4-  (coshv  sin?^  +  icosu  sinhv)-^^. 

What  is  the  meaning  of  the  i  which  occurs  in  these  scalar  func- 
tions ?     Is  the  magnitude  of  the  cosine 


or  IS  it 


V (cos  w  cosh  1;)^—  (sin^^sinhi?)^, 
cos  u  cosh  V  —  sinu  sinh  v  ? 


26      PRINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC  ANALYSIS. 

At  page  48  of  Definitions  of  the  Trigonometric  Functions,  I  show- 
that 

cos(w  +  iv)  =  -—^,    and   sin(w  -f-  iv)  =  -— ^j 

and  that  the  ordinary  proof  for  the  cosine  and  the  sine  of  the 
sum  of  two  angles  gives 

OK^  OM  ON     MP  NQ . 
OA      OA  OP      OA  OP ' 

that  is,  cos  {u  -j-  iv)  =  cos  u  cosh  v  —  sin  u  sinh  v, 

,  KQ^MP  ON     OMNQ . 

^^  OA      OA  OP      OA  OP' 

that  is,  sin  {u  -\-  iv)  =  sin  u  cosh  v  +  cos  u  sinh  v. 

What,  then,  is  the  function  of  the  i?  It  shows  that  if  you 
form  the  two  squares,  taking  account  of  it,  their  sum  will  be 
equal  to  unity.  Also,  in  forming  the  products  of  versors,  it  must 
be  taken  into  account.  When  it  is  preserved,  the  rules  for  cir- 
cular versors  apply  without  change  to  excircular  versors. 

Here  we  have  the  true  geometric  meaning  of  a  bi-versor,  and 
consequently  of  a  hi-quaternion ;  for  the  latter  is  only  the  former 
multiplied  by  a  line. 

As  a  special  case,  let  «^  =  ^  ;  we  then  have 

Li 

j8^  *"  =  —  i  sinh  v  -f  cosh  v-fi^  \ 
this  versor  evidently  refers  to  the  conjugate  hyperbola. 
Again,  let  %  =  tt  ;  we  have 

^TT+i.  =  —  (cosh  V  +  %  sinh-v  •  ^), 

which  refers  to  the  opposite  hyperbola. 

In  the  following  table,  the  related  excircular  versors  are  placed 
in  the  same  line  with  their  circular  analogues,  and  the  diagram 
(Fig.  14)  shows  the  related  versors  graphically. 


PRINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC  ANALYSIS.      27 


Circular. 

EXCIRCULAR. 

y8"       =      cosu  -{-sinu'p^ 

li'^           = 

coshw+isinhi^ 

•^^ 

AOPy 

^^  "   =      sin  u  -f  cos  u '  y8^ 

/-■"  = 

zsinhw+   coshw 

■P' 

AOP, 

5  +  M                                                                                          f 

y8^       =  — sinw4-cosz^./3 

n^*"  = 

—  zsinh?^4-  coshi* 

■P' 

AOP3 

/S'^-"    =—  cosu  +  smu-13^ 

;8"-'»      = 

—    coshu  +  Z  sinhi* 

■p' 

AOP, 

^■n+u      —  —  Q,o^u—&\l\U'fi^ 

^.+  .«      = 

-    coshz*— ^  sinh«t 

■P' 

AOP, 

/?  ^  "  =  — sini*  —  cosz^-y8^ 

li~^-"  = 

—  isinhit—  coshw 

■n^ 

AOP, 

/8          =      sm  2*  —  cos  l^  •  y8^ 

^-f+<»^ 

isinhw—  coshw 

•r 

AOP, 

yg-'*      =      COS  ^t  —  sin  z^ .  ^^ 

;8-'»     = 

coshii— i  sinhu 

■P' 

AOP, 

It  is  evident  that  AOP2  is  the  complement,  AOP^  the  supple- 
ment, and  AOP^  the  reciprocal,  of  AOPi.     It  is  not  the  circular 


Fig.  14. 


term  of  the  complex  exponent  which  is  affected  by  the  V— 1, 
but  the  excircular  term.  Thus  space  analysis  throws  a  new  light 
upon  the  periodicity  of  the  hyperbolic  functions. 


28      PRINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC  ANALYSIS. 


Fig.  15. 


,f+iu 


To  jind  the  product  of  tico  versors  of  the  equilateral  hyperholoid 
of  one  sheet,  when  each  passes  through  the  principal  axis  of  the 
hyperholoid. 

Let  P  be  a  point  on  the  excircle  of  one  sheet  (Fig.  15),  OP  its 
radius  J  draw  OB  equal  to  OA,  in  the  plane  of  OA  and  0P\   AB 

is  joined  by  a  quadrant  of  a  cir- 
cle, and  BOP  by  a  sector  of  an 
excircle.  Let  u  denote  the  ratio 
of  twice  the  area  of  the  sector 

POB  to  the  square  of  OA ;  ^  is 

the  ratio  of  twice  the  area  of 
BOA  to  the  square  of  OA. 
Hence  if  /8  is  a  unit  axis  per- 
pendicular to  OB  and  OA,  the 
expression  for  the  versor  POA 
is/S^^*".     Similarly,  the  expression  for  the  versor  AOQ  is  yi+'^ 

Now  )8^"^"*y^^"'  =  (  — isinhw+ cosh i^-j8^)(— ^■sinh'y-^ cosh -u-y^) 
=  —  (sinhw  sinhv  -f-  cosh?^  cosh-u  cos^Sy) 

—  f  i(cosh2t  sinhv  •  jB  -\-  coshv  sinh?i  •  y)  -f  cosh?^  cosht?  sin^y  •  a^. 

Now  the  magnitude  of  coshw  sinhv  •/? -f  cosh'y  sinhi^«y  may  be 
greater  or  less  than  cosh  ?^  cosh  v  sin  ^y.  If  it  is  greater,  then 
the  directed  sine  may  be  thrown  into  the  form 

—  i{  (cosh It  sinhv-ySH-  coshv  sinhw-y)  —  tcoshw  coshv  sin^y -aj, 

consequently,  the  ratio  is  excircular,  and  the  axis  excircular; 
hence  the  product  takes  the  form 

—  ^*"',  where  ^  =  cosh cfy-e  —  i  sinh (ft -a. 

But  if  coshw  cosh  V  sin /3y  is  the  greater,  the  directed  sine 
takes  the  form 

—  jcoshit  coshv  sin (Sy  a  -\- 1  (cosh  u  sinh'y-;8  4-  cosh  2;  sinhw-y)  j. 

The  ratio  of  the  product  is  circular,  but  the  axis  is  excircular. 
Let  w  denote  the  ratio ;  the  axis  has  the  form  cosh  <^  •  «  — i  sinh<^-e, 
so  that  the  product  is  of  the  form 

—  ^"'  =  —  cos^t7  —  sin !(;( cosh  <^  •  a  —  i  sinh<^  •  e)^. 


PRINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC  ANALYSIS.      29 

In  the  former  case,  the  locus  of  the  poles  of  the  axes  is  the 
exsphere  of  one  sheet;  in  the  latter,  the  opposite  sheet  of  the 
exsphere  of  two  sheets. 

To  find  the  product  of  two  general  versors  of  the  equilateral  hyper- 
holoid  of  one  sheet. 

The  one  versor  may  be  represented  by 

where  a?  —  y'^  -\-z^  =  1,   and   /8  is  perpendicular  to  a.     Similarly, 
the  other  versor  may  be  represented  by 

where  «'-  —  2/'^  4-  2;'^  =  1,  and  y  is  perpendicular  to  a. 

The  cosine  of  the  product  is 
xx'  H-  yy^  cos  ^y  —  2:2!', 

and  the  Sine  of  the  product  is 

i{xy^ '  y  +  x'y  •  ft)  +  (xz'  +  x'z  -\-  yy'  sinySy)  •  a. 

As  before,  if  (xy'y-{-(x'yy-\-2xx'yy'Gosfty  is  greater  than 
{xz'  +  x'z  +  yy'  sin  (SyY,  the  ratio  of  the  product  is  excircular ;  but 
if  less,  it  is  circular.  In  the  former  case  the  axis  is  an  axis 
of  the  exsphere  of  one  sheet,  in  the  latter  it  is  an  axis  of  the 
exsphere  of  two  sheets. 

To  find  the  product  of  two  versors  which  pass  through  the  prin- 
cipal axis,  when  the  one  belongs  to  the  exsphere  of  two  sheets,  the  other 
to  the  exsphere  of  one  sheet. 

Let  the  former  versor  be  denoted  by  j8^",  and  the  latter  by 
yf+ff.     Then 

"y^     =  (cosh  w  +  i  sinhw  •  l3^){  —  i  sinh-y+coshv  •  y^) 
=  —  I  (cosh  w  sinh-y  -f-  sinhw  cosh'j;  cos^y) 

n 

+  5 cosh u  cosh v-y  -\-  sinh u  sinh v-fS  —  i sinh u cosh v  sin j3y  -  a\ '\ 

As  the  magnitude  of  cosh?^  cosh v-yH- sinh w  sinh v-yS  is  by 
reasoning  similar  to  that  at  page  23  seen  to  be  greater  than 
sinh w  cosh v  sin ^y,  we  see  that  the  axis  is  excircular;  and  the 
i  before  the  scalar  term  shows  that  the  ratio  is  excircular.     From 


30      PKINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC  ANALYSIS. 

comparison  of  the  table,  page  27,  we  see  that  the  product  versor 
has  the  form 

^^^""j  where  $  —  cosh  <j>'€  —  i  sinh  <j>  • «, 
the  equations  being 

sinh  10  =  cosh  u  sinh  v  +  sinh  u  cosh  v  cos  /3y,  (1 ) 

cosh  w  sinh  <^  =  cosh  u  sinh  v  +  sinh  ?*  cosh  v  cos  jSy,  (2) 

cosh w cosh <ji'€  =  cosh i* cosh v •  y  +  sinh u  sinh v •  )8.  (3) 


FUNDAMENTAL   THEOREM   FOR   THE 
HYPBRBOLOID. 

The  theorems  for  the  hyperboloid  are  obtained  from  the  theo- 
rems for  the  exsphere  in  the  same  manner  as  the  theorems  for  the 
ellipsoid  are  deduced  from  those  for  the  sphere. 

Two  general  versors  for  the  hyperboloid  of  two  sheets  are 
expressed  by  ^'"  and  >;%  where 

^  =  cosh<^  (cosi//  -kp  -\-  sinj/^  -^V)  ~  ^'  sinh<^  •  a, 
and  -q  =  cosh  <j!>'  (cos  j/a'-  fc/3  +  sin  i//'-  h'y)  —  i  sinh  <^'-  a. 

Now     I*"  ry*"  =  (cosh  u  +  i  sinh  u-^^)  (cosh v  +  i  sinh v  -  rf^) 
=  cosh  u  cosh  v  +  sinh  u  sinh  ^'  cos  ^77 
H-5i(coshv  sinhw.^H- cosh M  sinh v.'>7)  +  sinh i*  sinh v  Sin|>;|   . 

The  problem  is  reduced  to  finding  the  versor  $r}.  We  apply  the 
same  principle  as  that  employed  in  finding  the  versor  between 
two  elliptic  axes  (page  13),  namely:  Restore  the  axes  to  their 
excircular  primitives,  find  the  versor  between  these  excircular 
axes  (page  23),  and  change  its  axis  according  to  the  ratios  of  the 
contraction  of  the  hyperboloid.     This  gives 

cos irj  =  cosh <^  cosh <f>'\GOs{\f/  —  \p')  \  —  sinh <^  sinh <^', 

Sin  ^-q  =  cosh  <^  cosh  <^'  sin  {ifz  —  ij/')  -a 

—  I  (cosh  <^  sinh  <f>'  sinxf/— cosh  <f>'  sinh  <^  sin  if/')  -k/B 
4-  i(cosh  <^  sinh  <^'  cos  i/^— cosh  <^'  sinh  <^  cos  ij/')  •  k'y. 

In  this  manner,  each  theorem  proved  for  the  exsphere  may  be 
generalized  for  the  hyperboloid. 


PRINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC  ANALYSIS.       31 


DB  MOIVRE'S  THEOREM. 

To  find  any  integral  power  of  a  versor. 

Let  n  denote  any  integral  nnmber.  For  the  general  spherical 
versor  we  have  (^")''=^*'",  because  the  axes  of  the  factor  versors 
are  all  the  same.     Hence 

cosnu-\- sin  nil- ^"^ 

=  (cos?*  + sinw'^  )** 

=  cos^'u  +  n  cos^^^i*  sinw .  ^^  -\ — ^— ^  cos**"^!*  sin^w  •  ^''+, 

from  which  it  follows  that 

cos  nu  =  cos"it  —  ^^  ^^  ~ — -  cos  "~^m  sinV  + , 

and  sin  nu  =  n  cos"~^w  sin  u  —  ^^ — ~  ^^ — — — -  eos'^'hi  sin^u  + . 

Similarly  for  the  exspherical  versor  (^'")",  as  the  axes  are  all 
the  same  (^■")"  =  ^*'*'',  and 

cosh72i^  +  i  sinhnw  •  ^^  =  (cosh?^  +  i  sinh  w  •  |  )'* 

=  cosh"w  +iii  cosh**"^?^  sinh  U'$^  -\ — ^^ -f  cosh'*~^w  sinh-w  •$''+', 

Zi  I 

therefore 

coshnw  =  cosh^w  +  ^^^~    ^cosh''~^i^  sinh^t*  +, 

2! 

and  sinh  wit  =  n  cosh'*~^t*  sinhi*  +  n\^n—   )\n—   )  qq^x-z^^y^u-\-. 

The  only  difference  in  the  case  of  the  general  ellipsoidal  versor 
is  that  u  is  measured  elliptically  and  ^  is  an  ellipsoidal  axis. 
So  for  the  general  hyperboloidal  versor,  u  is  measured  hyper- 
bolically  and  ^  is  a  hyperboloidal  axis. 


32      PRINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC  ANALYSIS. 

To  find  any  integral  root  of  a  versor. 

Consider  first  the  case  of  an  ellipsoidal  versor.     If  ii  is  defined 
as  the  ratio  of  twice  the  sector  to  the  rectangle  formed  by  the 

semi-axes,  it  cannot  be  greater  than  27r.     Then  (|")"  is  unambigu- 

u 

ously  equal  to  ^.     Hence 

cos-  +  sin-.|^  =  (cos?*  4-  sinw.p)«. 
n  n  ^  ^ 

If  cos  u  is  not  less  than  sin  u,  then 

u  u     "^  1  «■  L 

cos- +  sin -.^^  =  (cos%)"  \1  +  tanw.|^(« 

=  {co^uY  -j  1+  -tan I*  •  ^2  +  _L__Ztan2w  -^  -\-  [■ ; 


therefore 

cos 


^  =  (COS  I.)  U 1  +  i^^i^tan^.. 

_(n-l)(2n-l)(3»-l)^^^,        I 

.     .    u       .         xM^^  (n-l)(27?-l)^     ,        > 

and  sm-  =  (cosw)n  ■{  -tan it  —  ^^ ^V^— ^tan^i<  +  )-  - 

n      ^         ^    (n  n^Sl  > 

But  if  sin  u  is  not  less  than  cos  n,  we  have  the  complementary 
series 

«  i  J5L  _^   i 

|"  =  (sintt)"pJl-|-cotw.^  ^J«. 

Consider  next  the  case  of  a  hyperboloidal  versor.     A  versor  for 
the  hyperboloid  of  two  sheets  is  denoted  by  ^'■".     Now 

1  '-'*  IT      1 

(^■«)"  =  ^"  =  Jcoshtt  +  isinhw.pi" 

i  ir    i 

=  (cosh?*)";i  +  i  tanhw .  P  J% 

for  coshw  is  always  greater  than  sinhii ;  therefore 

cosh^  =  (coshw)^  1 1-  ^^f^ltanh^w 
n  (         71^  2 ! 

71*4!  ) 


PRINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC  ANALYSIS.      33 

(n-l)(2n-l) 


and  sinh-  =  (coshw)'*  \  -tanhi*  — 


n^Sl 


tanh^w+  I' 


But  a  versor  for  the  hyperboloid  of  one  sheet  is  expressed  by 
^^-^*".     Now 

(p+»")n  ^  p  '»  =  5  _  ^  sinh w  +  coshw .  PJ** 

=  (cosh  uyi^""  [l-i  tanh  u-f^}  % 

which  is  expanded  as  before. 


POLAR  THEOREM. 

To  deduce  in  the  trigonometry  of  the  sphere  the  polar  theorem, 
corresponding  to  the  fundamental  theorem. 

The  cosine  theorem,  which  is  the  fundamental  theorem  of 
spherical  trigonometry,  expresses  the  side  of  a  spherical  triangle 
in  terms  of  the  opposite  sides  and  their  included  angle.  In 
treatises  on  spherical  trigonometry,  it  is  shown  how  to  deduce 
from  the  cosine  theorem  a  polar  or  supple-  _ 

mental  theorem  which  expresses  an  angle 
in  terms  of  the  other  two  angles  and  the 
opposite  side.  It  is  our  object  to  find  the 
polar  theorem  corresponding  to  the  com- 
plete fundamental  theorem. 

Let  the  versors  of  the  three  sides  of  the 
spherical  triangle  (Fig.  16),  taken  the  same 
way  round,  be  denoted  by  |%  rf*,  ^"y  where 
$,  rj,  l  are  unit  axes,  and  a,  b,  c  denote  the 
ratio  of  twice  the  area  of  the  sector  to  the  area 
of  the  rectangle  formed  by  the  semi-axes  of  its  circle  (which,  in 
this  case,  is  simply  the  square  of  the  radius).  The  angles  in- 
cluded by  the  sides  are  usually  denominated  A,  B,  (7,  respectively, 
but  what  it  is  necessary  to  consider  in  view  of  further  generali- 
zation is  the  angles  between  the  planes,  or  rather  the  versors 
between  the  axes.  These  in  accordance  with  our  notation  are 
denoted  by  r^t,,  ^^,  and  ^yj  respectively ;  the  axes  of  these  versors, 
which  are  also  of  unit  length,  are  denoted  by  i;^,  ^|,  and  I17, 


Fig.  16. 


34      PKINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC  ANALYSIS. 

respectively,  and  they  correspond  to  the  poles  of  the  corners  of 
the  triangle  as  indicated  by  the  figure. 
The  fundamental  theorem  is 

^"r;*  =  cos  a  cos  b  —  sin  a  sin  b  cos  $rf 

n 

4-  [cos6  sina-^  +  cosa  sin6->;  —  sina  sin6  sin^iy.^ryp ; 
but  as  ^^  is  taken  in  the  opposite  direction,  we  have 
^^  =  cos  a  cos  b  —  sin  a  sin  b  cos  irj 

+  I  —  Gosb  sina-^—  cosa  sin  6  •  >y  +  sin  a  sin6  sin^iy -^r/p. 

The  polar  theorem  is  obtained  by  changing  each  side  into  the 
supplement  of  the  corresponding  angle  and  the  angle  into  the 
supplement  of  the  corresponding  side.     Hence 

cos  (tt  —  irj)  =  C0S(7r  —  y}^)  COS  (tt—  ^^) 

—  sin(7r  —  rjC)  sin(7r  —  ^^)  C0S(7r  —  c)  ; 
that  is,    cos  ^t;  =  —  cos  r}^  cos  ^$  —  sin  rj^  sin  ^$  cos  c. 

When  A,  B,  C,  are  used  to  denote  the  external  angles  between 
the  sides,  the  above  equation  is  written 

cos  C=  —  cos  A  cos  B  —  sin  A  sin  B  cos  c. 
Apply  the  same  rule  of  change  to  the  Sine  part,  and  we  obtain 

Sin(7r-^>y)  =  -COS(7r-^^)  Sin(7r-r;0 -COS(7r->70  Sin(7r-^^) 
+  sin(7r  —  r)^)  sin(7r  —  ^i)  sine •  ^ ; 

that  is,   Sin  ^t;  =  cos  ^i  Sin  rj^ + cos  rjC  Sin  ^| + sin  r)^  sin  ^$  sin  c  •  ^. 

To  deduce  the  polar  theorem  for  the  ellipsoid. 

Let  ^*,  if,  t,"  denote  the  three  versors  of  the  original  ellipsoidal 
triangle  taken  the  same  way  round;  then  the  corresponding 
versors  of  the  polar  triangle  are  i;^,  t,^,  and  ^rj.  The  third  versor 
of  the  original  triangle  is  given  in  terms  of  the  other  two  by  the 
theorem 

^'^  =  cos  a  cos  b  —  sin  a  sin  b  cos  ^rj 

TT 

-f  [  — cos6  sina- ^  — cosa  ^iwb-r)  +  sina  sin6  Sin^iy^. 


PRINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC  ANALYSIS.      35 

The  third  versor  of  the  polar  triangle  is  obtained  in  terms  of 
the  other  two  by  changing  each  versor  into  the  supplement  of  its 
corresponding  versor ;  hence 

cos  $r}  =  —  cos  r}^  COS  ^$  —  sin  ri^  sin  ^$  cos  c, 
and  Sin^i;  =  cos^^  Sinr;^  +  cos rj^  Sin^^  +  smrjt  sin^^  Sin^'. 

In  form  it  is  the  same  as  for  the  sphere ;  the  only  difference  is 
in  the  expressions  for  the  ellipsoidal  axes  i,  r),  C,  and  the  manner 
of  deducing  the  cosine  and  Sine  of  the  versor  between  two  such 
axes.  (See  page  13.)  The  polar  ellipsoid  is  not  identical  with 
the  original  ellipsoid;  the  ratios  of  the  two  minor  axes  are 
interchanged. 

To  deduce  the  polar  theorem  for  the  exsphere  of  two  sheets. 

Let  $'%  rf^,  l'"  denote  the  versors  for  the  three  sides  of  a  triangle 
of  the  exsphere  of  two  sheets,  taken  in  the  same  order  round. 
The  axes  i,  rj,  ^  have  their  poles  on  the  exsphere  of  two  sheets 
(page  23) ;  it  is  required  to  deduce  the  theorem  for  that  polar 
triangle.     For  the  original  triangle,  we  have 

^'«  =  cos  ia  cos  ib  —  sin  ia  sin  ib  cos  ir) 

IT 

+  \  —  COS  ib  sin  ia-i— cos  ia  sin  ib"q-\-  sin  ia  sin  ib  Sin  ^>;  J  ^. 

By  changing  each  versor  into  the  supplement  of  the  correspond- 
ing versor,  we  obtain 

^>y  =  —cosr;^  cos^l  —  siui;^  sin^^  coshc 

rr 

+  \(iOS^^  Sin?;^+cos?y^  Sin^l-f  I  sin^^^  sin^l  sinhc  •  ^p. 

The  above  cosine  equation  has  a  marked  resemblance  to  the 
fundamental  equation  of  non-euclidean  geometry  (see  Dr.  Giin- 
ther's  Hyperbelfunctionen,  pages  306  and  322).  It  is  true  that  77^ 
and  ^$  are  not  simple  circular  versors,  but  the  functions  are  cos 
and  sin  in  a  generalized  sense.  I  venture  the  opinion  that  non- 
euclidean  geometry  is  nothing  but  trigonometry  on  the  exsphere ; 
and  that  the  so-called  elliptic  and  hyperbolic  geometries  are  iden- 
tical with  the  ellipsoidal  and  hyperboloidal  trigonometry  developed 
in  this  paper. 


36      PEINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC  ANALYSIS. 

To  deduce  the  general  polar  theorem  for  the  exsphere. 

Let  $",  rf,  C  denote  the  three  sides  of  an  exspherical  triangle ; 
the  axes  ^,  rj,  ^  are  exspherical,  but  the  ratios  a,  b,  c  may  be  cir- 
cular or  excircular,  or  be  compounded  of  tt  or  f  and  an  excircular 
ratio.     For  the  original  triangle,  we  have 

^'  =  cos  a  cos  b  —  sin  a  sin  b  cos  iri 

-1-  J— cosa  sin6-^— cosa  sin6.7;  +  sina  sin6  Sin|?yj  , 

and  for  the  polar  triangle, 

^>y  =  —  cos  ry^  cos  ^^  —  sin  r;^  sin  ^^  cos  c 

+ 1  cos  C$  Sin  r]^ + cos  rj^  Sin  ^^ + sin  rjC  sin  ^^  Sin  ^^  p. 

Here  the  functions  cos  and  sin  are  used  in  their  most  general 
meaning. 

SINE  THEOREM. 

To  prove  that  if  $'^,  rf,  ^  denote  the  three  versors  of  a  spherical 
triangle,  then 

sin  rj^  _  sin^l^  _  sin  ^7; 
sin  a       sin  6       sine 

We  have  cose  =  cos  a  cos  6  —  sin  a  sin&  cos  ^77, 

and        sinc-^  =  —  cos  6  sin  a  •^— cos  a  sin  6  •  77  +  sin  a  sin  &  sin  ^17  •|>7. 

By  squaring  the  second  equation,  we  obtain 

sin^  c  =  cos^  b  sin^  a + cos^  a  sin^  b  4-  sin^  a  sin^  b  sin^  irj 
-f  2  cos  a  cos  b  sin  a  sin  b  cos  $ri ; 

then,  by  substituting  for  cos  $r)  from  the  first  equation,  and  reduc- 
ing, we  obtain 


sin  a  sin  b  sin  $r)  =  Vl  —  cos^  a — cos'^  6 — cos^  c + 2  cos  a  cos  b  cos  c. 

Hence  sin|2  ^  sin^  _  sin^^^ 

sin  c       sin  a       sin  b 

This  theorem  is  also  true  for  an  ellipsoid  of  revolution,  for  then 


sina  sin6sin^>7  =  A;  Vl  —  cos^a— cos^6— cos^c+2cosacos6cosc. 


PRINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC  ANALYSIS.      37 

To  find  the  analogue  for  the  exsphere  of  the  sine  theorem. 

Let  $,  rj,  ^  denote  exspherical  axes,  and  a,  b,  c  versors  which 
may  be  circular,  or  excircular,  or  both  combined.  Then,  with  the 
general  meaning  of  the  sin  and  cos  functions, 


sin  a  sin  6  sin  ^iy  =  Vl  —  cos^  a — cos^  6 — cos^  c + 2  cos  a  cos  6  cos  c. 

Hence  sinj,  ^  sin^^  ^  sin|| 

sine       sin  a       sin  6 

We  have  seen  that,  if  a  and  b  are  both  simply  excircular,  it 
does  not  follow  that  c  is  (page  28). 


SUM  AND  DIFFERENCE  THEOREMS. 

The  reciprocal  of  a  given  versor. 

By  the  reciprocal  of  a  given  versor  is  meant  the  versor  of 
equal  index  but  of  opposite  axis.  Let  ^"  denote  the  given 
spherical  versor;  its  reciprocal  is  (— |)".  But  it  may  be  shown 
that^"  =  (-^)^     For 

TT 

^-'*  =  cos(— i*)  +  sin(— w)  .^^ 
=  cosu  —  sinu'  i^ 

n 

=  cos  u  +  sin  u*  (  —  ^)  ^ 
=  (-«". 
Similarly  the  reciprocal  of  an  exspherical  versor  ^*"  is  (— 0*" 
or  ^~'",  and 

^-iu  _  coshw  —  i  sinhw  •  |  . 

The  reciprocal  of  an  ellipsoidal  versor  ^"  is  also  ^~",  the  only 
difference  being  that  |  is  no  longer  a  spherical,  but  an  ellipsoidal 
axis.     So  for  the  hyperboloidal  versor. 

To  find  the  analogues  of  the  sum  and  difference  theorems  of 
plane  trigonometry. 

At  page  45  of  "  The  Imaginary  of  Algebra,"  I  have  shown  how 
to  generalize  for  the  sphere  the  following  well-known  theorems 
in  plane  trigonometry,  namely, 

cos  (A-{-  B)-\-  cos  (A  —  B)=     2  cos  ^  cos  B, 

cos  (A  -\-  B)  —  GOs{A  —  B)  =  —  2  sin  A  sin  B, 


38      PRINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC  ANALYSIS. 

sin(^  +  -B)  +  sin(^  —  B)  =  2cos5  sin^, 
sin(J.-f  B)—  sin(^  —  B)=  2  cos  A  siiiB, 

and 


cos  (7  +  COS  Z>  = 

2cos^+^eos^;^, 

cos  0  —  cos  D  =  - 

-2sin<^  +  ^sin^;^, 

sin  C  4-  sin  Z>  = 

2sin— |-cos-^-, 

sin  C  —  sin  2>  = 

C+D    .    C-D 

2  cos  ^-^ —  sm      ^     ' 

The  generalized  formulae  of  the  first  set  for  the  sphere  are, 
using  general  axes  ^  and  -q, 

cos  ^7}^  +  cos  ^'^ry"^  =       2  cos  ^  COS  B, 

cos  ^^-q^  —  COS  ^^r)~^  =  —  2  COS  (Sin ^^  Sin  7;^), 
Sin  ^^yj^  +  Sin  ^^r;"^  =      2  cos  5  Sin  ^^ 

Sin^r;^  -  SinlV^  =      2  Jcos^  Sin,;^  -  Sin(Sin^^  Sin>;^)  J. 
Corresponding  to  the  latter  set  of  four  equations  we  have 

cos  r  +  cos  0)^  =       2C0S  Ja)^(a)-^^^)  ^1  COS((o-^n  '', 

cosC^-cos(o^  =  -2cos[SinJa)^(a>-^^^)^j  Sin(a>-^n4j 
Sin  ^^  +  Sin  0)^  =      2  cos  ( o)-^^^)  ^  Sin  \  o>^  (a>-^D '  \ , 
Sin^-  Sina)^=      2cos[o>^((o~^^^)^J  Sin(cu-^H^ 
-  2Sin  SinSa>^((o-^^^)^J  Sin((o-^ni 

The   corresponding  theorems   for  the  ellipsoid  are  the  same, 
excepting  that 

$  =  cos  cf} '  k/S  —  sin  (jt-a,     rj  =  cos  ({>'  -ky  —  sin  cf>'  -  a. 

Consequently  cos^rj  is  the  same  as  before,  but 
Sin^T;=  cos<^cos<^'sin/8ya— A;(cos<^sin<^'.^a  -\-cos(f>'  sin(^-ay). 

For  the  general  ellipsoid  the  only  difference  is  in  the  expres- 
sions for  $,  Y),  and  sin  ^rj-i-q. 


PRINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC  ANALYSIS.      39 


EXPONENTIAL  THEOREM. 

To  find  the  exponential  series  for  an  ellipsoidal  versor. 

In  the  expression  |"  for  a  spherical  versor,  the  u  and  i  are  truly 
related  as  index  to  base,  for  log^  =  ^l  log^^  =  u .  ^^,  and  therefore 
^  =  e"'^^.     Consequently 

2!      ^4! 


*{'-f,-f,-}-^' 


In  the  case  of  the  spherical  versor,  ^  =  cos  <j^  • /?  —  sin  <^  •  ce,  or 
cos  <^  (cos  j/^  •  /?  +  sin  j/^  •  y)  —  sin  <^  •  a,  where  a,  (3,  y  are  unit  axes 
mutually  rectangular. 

The  expansion  for  the  ellipsoidal  versor  ^  differs  only  in  the 
way  in  which  u  is  measured,  and  in  the  expression  for  ^,  which  is 
now  GOS<f>-k^  —  sin<^.a,  or  cos<^(cosj/'- A;y8+  smKf/'Jc'y)  —  sin </>•«. 

To  find  the  exponential  series  for  a  hyperholoidal  versor. 

The  expression  for  a  versor  on  the  exsphere  of  two  sheets  is 
^\     Kow 


ir    .  ii: 


=  1+—+  —  + 

2!      4! 
The  expression  for  a  mixed  exspherical  versor  is  ^"'■*"''.     Now 

tu-\-iV  __   g(M+it)).ff 

=  i+(„  +  fo).^i+(!L±M!.f.  +  Oi±M!./i  + 

2!        "^        4! 


,    (      ,    .         (u-{-ivY  ,       )     A 


40      PRINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC  ANALYSIS. 

Both  the  cosine  and  the  sine  break  up  into  two  components,  the 
one  independent  of  t,  and  the  other  involving  i.  Here  we  have 
the  sine  and  the  cosine  of  the  ordinary  complex  quantity. 

As  the  ratio  of  a  hyperboloidal  versor  may  be  circular  or  excir- 
cular,  or  both  combined,  the  general  versor  may  be  expressed  by 
^*,  where  a  is  as  general  as  stated.     Then 

2!      4! 


To  find  the  exponential  series  for  the  product  of  two  ellipsoidal 
versor  s. 

In  the  paper  on  The  Fundamental  Theorems  of  Analysis  Gener- 
alized for  Space  I  have  shown  that  if  ^"  and  rf  denote  any  two 
spherical  versors,  then 


^ry"  =  e"-^^+''"''^ 


.^\  _1_  J-    /^o,-i^_L^,-^2\2    I       1    /„,  .  t'i     !    ...  _2\3. 


where  the  powers  of  the  binomial  are  expanded  according  to  the 
binomial  theorem,  but  subject  to  the  special  proviso  that  the  order 
of  the  axes  |,  -q  must  be  preserved  in  all  the  axial  terms.     Thus 

^rf^l-^U^e^V'-q^ 

o ! 
4- etc. 


=  l  —  —  \u^-\-2uV  COS^r;  +  ^^5 

Zi  I 

4-  —  S^t'*  +  4w^u  cos ^7/  -f  6wV+4wv^cos|?;4-'u^i 
—  etc. 


(1) 


PRINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC  ANALYSIS.      41 

+  I  ^  _  |.^  (3i^2^  +  v')  +  etc.  I .  -n^  (3) 

-j_  j  _  J_  2  uv  +  -^  (4 w^^  +  4  i^v"^)  -  etc.  |  sin ^>;  •  ?^^.      (4) 

In  the  case  of  the  sphere 

^  =  cos  <f>'  p  —  sin  <^  •  a, 
and  7]  =  cos  <f>'  -y  —  sin  <;t'  •  a ; 

consequently  cos  ^>y  =  cos  </>  cos  <^'  cos  fSy  +  sin  <^  sin  <j>',  and 
Sin  It;  =  cos  cj>  cos  <^'  sin  jSy-a  —  (cos  <^  sin  cf>'  -  13a  -\-  cos  <^'  sin  <^  •  ay ) . 

For  the  ellipsoid  of  revolution  the  expansion  is  obtained  by 
introducing  ellipsoidal  axes  $  and  tj  ;  and  the  corresponding  theo- 
rems for  the  hyperboloid  are  obtained  by  changing  the  axes  and 
indices  into  hyperboloid  axes  and  indices. 

To  find  the  exponential  series  for  the  product  of  two  hyperboloidal 
versors. 

Let  i  and  rj  denote  any  two  hyperboloidal  axes,  and  u  and  v 
general  hyperboloidal  ratios  (p.  40) .     Then  the  product  is 

The  form  of  the  theorem  is  the  same  as  before. 


LOGARITHMIC  VERSORS. 

In  the  paper  on  The  Fundamental  Theore^ns  of  Analysis  Gen- 
eralized for  Space,  page  16,  I  have  shown  that  when  the  index  of 
a,  in  e"^*^,  is  generalized,  we  obtain  the  expression  for  the  versor 


42      PKINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC  ANALYSIS. 

corresponding  to  a  sector  of  a  logarithmic  spiral.     Let  w  denote 
the  general  angle,  and  a^  the  generalized  versor ;  then 


=  1  +  ^  cos  w  H — 1 -^ h  etc. 

(    .    .         ,  A^sm2w  ,  J.^sin3w  ,   ^. ^  )       f 


4-  I  ^  sin  w  H — h ~ +  etc 

nA  COS  w  aA  sin  w .  a^ 


V 


It  is  there  shown  that  w  is  the  constant  angle  between  the  radius, 
vector  and  the  tangent,  or  rather  that  it  is  the  constant  difference 
between  the  circular  versor  from  the  principal  axis  to  the  tan- 
gent, and  that  from  the  principal  axis  to  the  radius  vector.  It  is 
also  shown  that  A  sinw  gives  the  ratio  of  twice  the  area  of  the 
corresponding  circular  sector  to  the  square  of  the  radius,  while 
A  cos  w  gives  the  logarithm  of  the  ratio  of  the  radius  vector  to 
the  principal  axis. 

I  have  there  called  such  a  logarithmic  versor,  when  multiplied 
by  a  length,  a  quinternion.  In  his  Synopsis  der  Hoheren  Mathe- 
matik,  Mr.  Hagen  has  pointed  out  that  the  proper  classical  word 
is  quinion.  A  quaternion  means  a  ratio  of  three  elements  mul- 
tiplied by  a  length ;  therefore,  a  ratio  involving  an  additional 
element  when  multiplied  by  a  length,  is  a  quinion. 

In  the  paper  on  The  Imaginary  of  Algebra,  an  excircular  ana- 
logue is  deduced,  namely,  af^  =  e^*'"",  but  there  are  in  reality 
three,  according  to  whether  A  or  w,  or  both,  are  affected  by  the 

To  deduce  the  four  forms  of  logarithmic  versor. 
First:    circular-circular.      Let   ^"   denote   a  general   spherical 
versor,  then 

dM  __  /3»*|^  ^  pu  COS  tv+u  Bin  w^  2 

==l  +  ^^««4.|!^2._^|!^^3.^etc. 

Here  w  denotes  the  constant  difference  between  the  versor  from 
the  principal  axis  to  the  tangent  and  that  from  the  principal 
axis  to  the  radius  vector. 


PRINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC  ANALYSIS.      43 

Second:  circular-excircular.  Let  iw  denote  the  constant  dif- 
ference between  the  excircular  versor  from  the  principal  axis  to 
the  tangent,  and  that  from  the  principal  axis  to  the  radius 
vector;  then 

2  3 

=  1  -t-  wcosh^y-^-— cosh2io  +  — cosh  Sty  + 
2 !  3 ! 

-\-i^u  sinhw  -f  —  sinh2 w  +  ^  sinhSw;  +  |  •  ^^. 

Third :  excircular-circular.    Let  ^'"  denote  a  general  exspherical 

versor ;  it  is  equal  to  e***"^^,  and  here  f  denotes  the  constant  sum     , 
of  the  circular  versors  above  mentioned.     Let  that  constant  sum 
be  any  other  circular  versor  w.     Then 

IT 

tiu  __  piu  -I'"  __  piu  cos  w+iu  sin  w  •  ^'2 

=  l  +  ^^.^-  +  iM.^^2._^i|I^^3.  +  etc. 

^ !  ol 

2!  4! 

=  1  —  —  cos  2  w  +  —  COS  4  w  —  etc. 
■i-i  -lu  cos  10 cos  3w-\-  etc.  >- 

+   |_^sin2«;  +  ^sin4w;-|-|^ 
(2!  4!  ) 

+  i  ■]  w  sin  w  —  ^  sin  3  w  +  etc.  >■  •  $^. 

Here  both  the  cosine  and  the  sine  consists  of  a  real  and  an  ap- 
parently imaginary  part.  The  geometrical  meaning  has  already 
been  explained  (page  25). 


44      PRINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC  ANALYSIS. 

Fourth :   excircular-excircular.     Let  iw  denote  the  constant  sum 
of  the  excircular  versors  mentioned  in  the  second  case.     Then 

tiu  __  giM  •  f*'"  __  gtMCOshw-Msinhtc  •  ^^ 

=  1  +  fo .  i" + ^  •  e"  +  ^  •  e'"  + 

=  1  -  ^(oosh  2W  +  i  sinh  2«)  •  i^)  + 

4-  m(cosh  w  +  i  sinhw  •  ^)  — 
=  1  —  — -  cosh  2  w  +  ^  cosh  4  w  — 

■\-i\n  cosh  w  —  —  cosh  ^w  -\-  )■ 

—    -j  w  sinh  w  —  —  sinh  3^4-  ^-  •  ^^ 

-\-^\-~  sinh  2  w  +  f-  sinh  4t(;  -  |  •  ^^. 


To  find  the  product  of  two  logarithmic  versors  of  the  most  general 
kind. 

Let  ^  and  r}  denote  general  axes,  and  u,  w,  v,  t  general  ratios ; 
that  is,  each  may  be  a  sum  of  a  circular  and  an  excircular  ratio. 
Then  $1  and  rjt  each  denote  a  general  logarithmic  versor.     Then 

The  powers  of  the  binomial  are  formed  according  to  the  same 
rule  as  before.     (Fundamental  Theorems,  page  18.) 


COMPOSITION   OP   ROTATIONS. 

To  find  the  resultant  of  two  elliptic  rotations  round  axes  which 
pass  through  a  common  point. 

Two  circular  rotations  are  compounded  by  the  principle  that 
the  product  of  the  half  rotations  is  half  of  the  resultant  rotation. 


PRINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC  ANALYSIS.      45 

Let  any  two  circular  rotations  be  denoted  by  ^"  and  rf,  and 
their  resultant  by  I'*  x  ry  ;  then 

=  \  COS  -  COS sm  -  sm  -  cos  ^w 

+  /'cos^sin  I  •  ^  +  cos  I  sin^ . ,;  -  sin  |  sin|  Sin^^V  |-  '• 

Let  a;  =  cos  -  cos  -  —  sm  -  sin  -  cos  ^rj, 

2        2  2^ 


2/  =  Vl  —  a^, 

COS-  sin-'|4-cos-sin-«w  —  sm-  sm-  ISm^w 
2       2  2       2^  22  \ 

then  ^«  X  •»?"  =  a;2  -  2/2  +  2a;2/ •  ^^. 

The  elliptic  generalization  is  obtained  by  generalizing  the  axes 
i  and  rj  and  finding  cos^ry  and  Sin^iy,  as  at  page  15. 

To  find  the  resultant  of  two  hyperbolic  rotations  round  axes  which 
pass  through  a  common  point. 

Let  ^"^  and  rf"  denote  two  exspherical  rotations  which  have  a 
common  principal  axis ;  let  their  resultant  be  denoted  by  ^*"  x  ly*". 
By  analogy  we  deduce  that 

iu   iv 

=  \  cosh  ^  cosh  -  +  sinh  -  sinh  -  cos  in 
(  2  2  2         2' 

+  /cosh  I  sinh  ^  •  ^  +  cosh-  sinh^  -rj-i  sinh^ sinh | Sin  irj^  I  \ 
\         2  2  2  2  2  2  J    ) 

Let  X  =  cosh  ^  cosh  |  +  sinh  ^  sinh  ^  cos  ^. 


cosh-  sinh-  •  ^+cosh-  sinh-  >  n—i  sinh-  sinh-  Sin ^77 
^_  2         2  2  2     ' 22' 

Then  p  X  ^  =  a^  +  2/'  +  2  a;2/  •  ^^. 


46      PRINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC  ANALYSIS. 

Suppose  a  fluid  to  move  round  the  axis  i,  each  particle  describ- 
ing a  hyperbolic  angle  u,  and  then  round  the  axis  17  by  a  hyper- 
bolic angle  v,  the  principal  axes  of  the  two  motions  coinciding ; 
the  resultant  gives  the  angle,  the  plane,  and  the  principal  axes  of 
the  equivalent  single  motion  of  the  same  kind.  The  axis  of  that 
motion  does  not  pass  through  the  intersection  of  the  axes  of  the 
components. 

A  more  general  result  is  obtained  by  supposing  the  ratios  to 
be  complex;  the  theorem  is  then  expressed  by  the  spherical 
theorem  taken  in  a  generalized  sense,  just  as  in  ordinary  algebra 
a;  may  be  positive  or  negative. 

To  find  the  effect  of  an  elliptic  rotation  on  a  line. 

The  effect  of  a  circular  rotation  ^"  upon  a  unit  axis  p,  is  given 
by  the  equation 

^"p  =  cos^/o  •  ^  +  sin?*  Sin^/o  +  cos  w  Sin(Sin^/o)^. 
{Principles  of  the  Algebra  of  Physics,  page  100.) 

It  was  shown  by  Cayley  that  the  effect  of  ^"  upon  p  is  given 
by  the  Sine  of  the  product  ^  '^  p^  $^.     For  by  the  expansion  of 

(cos^_sin|.,y(coB|  +  sin|.,^) 

the  directed  sine  is  found  to  be 

cos^- .  p  +  sin^^cos ^p  •  ^  +  cos^  sin|  Sin ip  -  sin^^  Sin(Sin^p)^. 

But      cos^^.  p  =  cos^l  cos^p .  ^  +  cos2|Sin(Sin  ^p)|, 

therefore  the  directed  sine  is 

cos  ^p  •  ^  -I-  sin  w  Sin  ip  +  cos  u  Sin  (Sin  ip)  |. 

To  generalize  for  an  elliptic  rotation  we  substitute  the  more 
general  value  of  i  and  form  cos  |p,  Sin^p,  and  Sin  (Sin  ^p)^,  accord- 
ing to  the  rules  stated  at  page  15.     For  example,  let 

^  =  fc  cos  <^  •  ^  —  sin  <^  •  a, 
p  =  sin  ^  •  y  +  cos  ^  •  a ; 


PRINCIPLES  OF  ELLIPTIC  AND  HYPERBOLIC  ANALYSIS.      47 


then 

cos  $p  =  cos  0  sin  6  cos  fty  —  sin  <^  cos  0, 

Sin^p  =  cos <^  sin ^  sin/3y  •  a  +  A:(cos<^cos0- /Set  —  sin</)sin^.ay). 

To  find  the  effect  of  a  hyperbolic  rotation  on  a  line. 

Consider  the  simplest  exspherical  analogue  of  the  spherical 
theorem  of  the  preceding  article ;  it  is 

pp  =  cos^/o  •  ^  +  i  sinhw  Sin^p  -|-  coshi*  Sin  (Sin  ^/o)^. 
But  ^  is  now  an  excircular  axis  of  the  form 

^  =  cosh  <f>'  /3  —  i  sinh  <^  •  a. 
Let,  as  before,  p  =  sin  ^  •  y  4-  cos^  •  a ; 
then  cos^/o  =  cosh<^  sin^  cos^y  —  i  sinh  <^  cos  ^, 
Sin  ip  =  cosh  <^  sin  ^  sin  )8y  •  a  +  cosh  <^  cos  6-  fia—  i  sinh  <^  sin  ^  •  ay, 
Sin  (Sin  ^p)^ 

=  cosh^  <^  sin  6  sin  /3y  •  «y8  +  cosh^  <^  cos  ^  •  ct  —  sinh^  <^  sin  ^  •  y 
—  i  cosh  <^  sinh  <^  cos  O-ft—i  cosh  <;^  sinh  <^  sin  B  sin  ayfi  •  a. 

The  effect  of  a  hyperbolic  rotation  is  obtained  by  taking  the 
more  general  value  of  ^  and  applying  the  hyperbolic  rules  of 
multiplication. 


■^1^^. 


M''-'^ 


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PAPEKS   OK   SPACE  ANALYSIS 

By  PROFESSOR    MACFARLANE 

1.  Principles  of  tlie  Algebra  of  Physics     .    .  $1.00 

2.  The  Imaginary  of  Algebra 50 

3.  The   Fundamental   Theorems   of  Analysis 

Generalized  for  Space 50 

4.  On  the   Definitions  of    the  Trigonometric 

Functions 50 

5.  The  Principles  of  Elliptic  and  Hyperbolic 

Analysis 50 

■MAY     BE    OBTAINED    FROM 

B.    WESTERMAXN    &    CO., 

812  Broadway,  New  York. 

MA(\MILLAN   &   BOWES, 

Cambridge,  England. 

A.    HERMANN, 

8  Rue  de  la  Sorbonne,  Paris,  France. 

R.    FRIED  LANDER   &   SOHN,  * 

Carlstrasse  U,  Berlin,  Germany. 

THE    AUTHOR, 

University  of  Texas,  Austin,  'i'exas. 


